Danh sách tích phân với hàm lượng giác

Lượng giác

• Khái quát
• Lịch sử
• Ứng dụng

• Hàm
  • Hàm ngược

Tham khảo

• Đẳng thức
• Giá trị đặc biệt
• Bảng
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Định lý

• Sin
• Cos
• Tang
• Cotang

• Pythagoras

Vi tích phân

• Phép thế lượng giác
• Tích phân
  • Hàm nghịch đảo
• Đạo hàm

Đây là danh sách tích phân (nguyên hàm) của các hàm lượng giác. Đối với tích phân của chứa hàm lượng giác và hàm mũ, xem Danh sách tích phân với hàm mũ. Đối với danh sách đầy đủ các tích phân, xem Danh sách tích phân. Đối với danh sách các tích phân đặc biệt của các hàm lượng giác, xem Tích phân lượng giác.

Nhìn chung, với cos ⁡ ( x ) {displaystyle cos(x)} là đạo hàm của hàm số sin ⁡ ( x ) {displaystyle sin(x)} , ta có

∫ a cos ⁡ n x d x = a n sin ⁡ n x + C {displaystyle int acos nx,dx={frac {a}{n}}sin nx+C}

Trong mọi công thức dưới đây, a là một hằng số khác không và C ký hiệu cho hằng số tích phân.

∫ sin ⁡ a x d x = − 1 a cos ⁡ a x + C {displaystyle int sin ax,dx=-{frac {1}{a}}cos ax+C} ∫ sin 2 ⁡ a x d x = x 2 − 1 4 a sin ⁡ 2 a x + C = x 2 − 1 2 a sin ⁡ a x cos ⁡ a x + C {displaystyle int sin ^{2}{ax},dx={frac {x}{2}}-{frac {1}{4a}}sin 2ax+C={frac {x}{2}}-{frac {1}{2a}}sin axcos ax+C} ∫ sin 3 ⁡ a x d x = cos ⁡ 3 a x 12 a − 3 cos ⁡ a x 4 a + C {displaystyle int sin ^{3}{ax},dx={frac {cos 3ax}{12a}}-{frac {3cos ax}{4a}}+C} ∫ x sin 2 ⁡ a x d x = x 2 4 − x 4 a sin ⁡ 2 a x − 1 8 a 2 cos ⁡ 2 a x + C {displaystyle int xsin ^{2}{ax},dx={frac {x^{2}}{4}}-{frac {x}{4a}}sin 2ax-{frac {1}{8a^{2}}}cos 2ax+C} ∫ x 2 sin 2 ⁡ a x d x = x 3 6 − ( x 2 4 a − 1 8 a 3 ) sin ⁡ 2 a x − x 4 a 2 cos ⁡ 2 a x + C {displaystyle int x^{2}sin ^{2}{ax},dx={frac {x^{3}}{6}}-left({frac {x^{2}}{4a}}-{frac {1}{8a^{3}}}right)sin 2ax-{frac {x}{4a^{2}}}cos 2ax+C} ∫ x sin ⁡ a x d x = sin ⁡ a x a 2 − x cos ⁡ a x a + C {displaystyle int xsin ax,dx={frac {sin ax}{a^{2}}}-{frac {xcos ax}{a}}+C} ∫ ( sin ⁡ b 1 x ) ( sin ⁡ b 2 x ) d x = sin ⁡ ( ( b 2 − b 1 ) x ) 2 ( b 2 − b 1 ) − sin ⁡ ( ( b 1 + b 2 ) x ) 2 ( b 1 + b 2 ) + C ( | b 1 | ≠ | b 2 | ) {displaystyle int (sin b_{1}x)(sin b_{2}x),dx={frac {sin((b_{2}-b_{1})x)}{2(b_{2}-b_{1})}}-{frac {sin((b_{1}+b_{2})x)}{2(b_{1}+b_{2})}}+Cqquad {mbox{(}}|b_{1}|neq |b_{2}|{mbox{)}}} ∫ sin n ⁡ a x d x = − sin n − 1 ⁡ a x cos ⁡ a x n a + n − 1 n ∫ sin n − 2 ⁡ a x d x ( n > 0 ) {displaystyle int sin ^{n}{ax},dx=-{frac {sin ^{n-1}axcos ax}{na}}+{frac {n-1}{n}}int sin ^{n-2}ax,dxqquad {mbox{(}}n>0{mbox{)}}} ∫ d x sin ⁡ a x = − 1 a ln ⁡ | csc ⁡ a x + cot ⁡ a x | + C {displaystyle int {frac {dx}{sin ax}}=-{frac {1}{a}}ln {left|csc {ax}+cot {ax}right|}+C} ∫ d x sin n ⁡ a x = cos ⁡ a x a ( 1 − n ) sin n − 1 ⁡ a x + n − 2 n − 1 ∫ d x sin n − 2 ⁡ a x ( n > 1 ) {displaystyle int {frac {dx}{sin ^{n}ax}}={frac {cos ax}{a(1-n)sin ^{n-1}ax}}+{frac {n-2}{n-1}}int {frac {dx}{sin ^{n-2}ax}}qquad {mbox{(}}n>1{mbox{)}}} ∫ x n sin ⁡ a x d x = − x n a cos ⁡ a x + n a ∫ x n − 1 cos ⁡ a x d x = ∑ k = 0 2 k ≤ n ( − 1 ) k + 1 x n − 2 k a 1 + 2 k n ! ( n − 2 k ) ! cos ⁡ a x + ∑ k = 0 2 k + 1 ≤ n ( − 1 ) k x n − 1 − 2 k a 2 + 2 k n ! ( n − 2 k − 1 ) ! sin ⁡ a x = − ∑ k = 0 n x n − k a 1 + k n ! ( n − k ) ! cos ⁡ ( a x + k π 2 ) ( n > 0 ) {displaystyle {begin{aligned}int x^{n}sin ax,dx&=-{frac {x^{n}}{a}}cos ax+{frac {n}{a}}int x^{n-1}cos ax,dx&=sum _{k=0}^{2kleq n}(-1)^{k+1}{frac {x^{n-2k}}{a^{1+2k}}}{frac {n!}{(n-2k)!}}cos ax+sum _{k=0}^{2k+1leq n}(-1)^{k}{frac {x^{n-1-2k}}{a^{2+2k}}}{frac {n!}{(n-2k-1)!}}sin ax&=-sum _{k=0}^{n}{frac {x^{n-k}}{a^{1+k}}}{frac {n!}{(n-k)!}}cos left(ax+k{frac {pi }{2}}right)qquad {mbox{(}}n>0{mbox{)}}end{aligned}}} ∫ sin ⁡ a x x d x = ∑ n = 0 ∞ ( − 1 ) n ( a x ) 2 n + 1 ( 2 n + 1 ) ⋅ ( 2 n + 1 ) ! + C {displaystyle int {frac {sin ax}{x}},dx=sum _{n=0}^{infty }(-1)^{n}{frac {(ax)^{2n+1}}{(2n+1)cdot (2n+1)!}}+C} ∫ sin ⁡ a x x n d x = − sin ⁡ a x ( n − 1 ) x n − 1 + a n − 1 ∫ cos ⁡ a x x n − 1 d x {displaystyle int {frac {sin ax}{x^{n}}},dx=-{frac {sin ax}{(n-1)x^{n-1}}}+{frac {a}{n-1}}int {frac {cos ax}{x^{n-1}}},dx} ∫ d x 1 ± sin ⁡ a x = 1 a tan ⁡ ( a x 2 ∓ π 4 ) + C {displaystyle int {frac {dx}{1pm sin ax}}={frac {1}{a}}tan left({frac {ax}{2}}mp {frac {pi }{4}}right)+C} ∫ x d x 1 + sin ⁡ a x = x a tan ⁡ ( a x 2 − π 4 ) + 2 a 2 ln ⁡ | cos ⁡ ( a x 2 − π 4 ) | + C {displaystyle int {frac {x,dx}{1+sin ax}}={frac {x}{a}}tan left({frac {ax}{2}}-{frac {pi }{4}}right)+{frac {2}{a^{2}}}ln left|cos left({frac {ax}{2}}-{frac {pi }{4}}right)right|+C} ∫ x d x 1 − sin ⁡ a x = x a cot ⁡ ( π 4 − a x 2 ) + 2 a 2 ln ⁡ | sin ⁡ ( π 4 − a x 2 ) | + C {displaystyle int {frac {x,dx}{1-sin ax}}={frac {x}{a}}cot left({frac {pi }{4}}-{frac {ax}{2}}right)+{frac {2}{a^{2}}}ln left|sin left({frac {pi }{4}}-{frac {ax}{2}}right)right|+C} ∫ sin ⁡ a x d x 1 ± sin ⁡ a x = ± x + 1 a tan ⁡ ( π 4 ∓ a x 2 ) + C {displaystyle int {frac {sin ax,dx}{1pm sin ax}}=pm x+{frac {1}{a}}tan left({frac {pi }{4}}mp {frac {ax}{2}}right)+C} ∫ cos ⁡ a x d x = 1 a sin ⁡ a x + C {displaystyle int cos ax,dx={frac {1}{a}}sin ax+C} ∫ cos 2 ⁡ a x d x = x 2 + 1 4 a sin ⁡ 2 a x + C = x 2 + 1 2 a sin ⁡ a x cos ⁡ a x + C {displaystyle int cos ^{2}{ax},dx={frac {x}{2}}+{frac {1}{4a}}sin 2ax+C={frac {x}{2}}+{frac {1}{2a}}sin axcos ax+C} ∫ cos n ⁡ a x d x = cos n − 1 ⁡ a x sin ⁡ a x n a + n − 1 n ∫ cos n − 2 ⁡ a x d x ( n > 0 ) {displaystyle int cos ^{n}ax,dx={frac {cos ^{n-1}axsin ax}{na}}+{frac {n-1}{n}}int cos ^{n-2}ax,dxqquad {mbox{(}}n>0{mbox{)}}} ∫ x cos ⁡ a x d x = cos ⁡ a x a 2 + x sin ⁡ a x a + C {displaystyle int xcos ax,dx={frac {cos ax}{a^{2}}}+{frac {xsin ax}{a}}+C} ∫ x 2 cos 2 ⁡ a x d x = x 3 6 + ( x 2 4 a − 1 8 a 3 ) sin ⁡ 2 a x + x 4 a 2 cos ⁡ 2 a x + C {displaystyle int x^{2}cos ^{2}{ax},dx={frac {x^{3}}{6}}+left({frac {x^{2}}{4a}}-{frac {1}{8a^{3}}}right)sin 2ax+{frac {x}{4a^{2}}}cos 2ax+C} ∫ x n cos ⁡ a x d x = x n sin ⁡ a x a − n a ∫ x n − 1 sin ⁡ a x d x = ∑ k = 0 2 k + 1 ≤ n ( − 1 ) k x n − 2 k − 1 a 2 + 2 k n ! ( n − 2 k − 1 ) ! cos ⁡ a x + ∑ k = 0 2 k ≤ n ( − 1 ) k x n − 2 k a 1 + 2 k n ! ( n − 2 k ) ! sin ⁡ a x = ∑ k = 0 n ( − 1 ) ⌊ k / 2 ⌋ x n − k a 1 + k n ! ( n − k ) ! cos ⁡ ( a x − ( − 1 ) k + 1 2 π 2 ) = ∑ k = 0 n x n − k a 1 + k n ! ( n − k ) ! sin ⁡ ( a x + k π 2 ) ( n > 0 ) {displaystyle {begin{aligned}int x^{n}cos ax,dx&={frac {x^{n}sin ax}{a}}-{frac {n}{a}}int x^{n-1}sin ax,dx&=sum _{k=0}^{2k+1leq n}(-1)^{k}{frac {x^{n-2k-1}}{a^{2+2k}}}{frac {n!}{(n-2k-1)!}}cos ax+sum _{k=0}^{2kleq n}(-1)^{k}{frac {x^{n-2k}}{a^{1+2k}}}{frac {n!}{(n-2k)!}}sin ax&=sum _{k=0}^{n}(-1)^{lfloor k/2rfloor }{frac {x^{n-k}}{a^{1+k}}}{frac {n!}{(n-k)!}}cos left(ax-{frac {(-1)^{k}+1}{2}}{frac {pi }{2}}right)&=sum _{k=0}^{n}{frac {x^{n-k}}{a^{1+k}}}{frac {n!}{(n-k)!}}sin left(ax+k{frac {pi }{2}}right)qquad {mbox{(}}n>0{mbox{)}}end{aligned}}} ∫ cos ⁡ a x x d x = ln ⁡ | a x | + ∑ k = 1 ∞ ( − 1 ) k ( a x ) 2 k 2 k ⋅ ( 2 k ) ! + C {displaystyle int {frac {cos ax}{x}},dx=ln |ax|+sum _{k=1}^{infty }(-1)^{k}{frac {(ax)^{2k}}{2kcdot (2k)!}}+C} ∫ cos ⁡ a x x n d x = − cos ⁡ a x ( n − 1 ) x n − 1 − a n − 1 ∫ sin ⁡ a x x n − 1 d x ( n ≠ 1 ) {displaystyle int {frac {cos ax}{x^{n}}},dx=-{frac {cos ax}{(n-1)x^{n-1}}}-{frac {a}{n-1}}int {frac {sin ax}{x^{n-1}}},dxqquad {mbox{(}}nneq 1{mbox{)}}} ∫ d x cos ⁡ a x = 1 a ln ⁡ | tan ⁡ ( a x 2 + π 4 ) | + C {displaystyle int {frac {dx}{cos ax}}={frac {1}{a}}ln left|tan left({frac {ax}{2}}+{frac {pi }{4}}right)right|+C} ∫ d x cos n ⁡ a x = sin ⁡ a x a ( n − 1 ) cos n − 1 ⁡ a x + n − 2 n − 1 ∫ d x cos n − 2 ⁡ a x ( n > 1 ) {displaystyle int {frac {dx}{cos ^{n}ax}}={frac {sin ax}{a(n-1)cos ^{n-1}ax}}+{frac {n-2}{n-1}}int {frac {dx}{cos ^{n-2}ax}}qquad {mbox{(}}n>1{mbox{)}}} ∫ d x 1 + cos ⁡ a x = 1 a tan ⁡ a x 2 + C {displaystyle int {frac {dx}{1+cos ax}}={frac {1}{a}}tan {frac {ax}{2}}+C} ∫ d x 1 − cos ⁡ a x = − 1 a cot ⁡ a x 2 + C {displaystyle int {frac {dx}{1-cos ax}}=-{frac {1}{a}}cot {frac {ax}{2}}+C} ∫ x d x 1 + cos ⁡ a x = x a tan ⁡ a x 2 + 2 a 2 ln ⁡ | cos ⁡ a x 2 | + C {displaystyle int {frac {x,dx}{1+cos ax}}={frac {x}{a}}tan {frac {ax}{2}}+{frac {2}{a^{2}}}ln left|cos {frac {ax}{2}}right|+C} ∫ x d x 1 − cos ⁡ a x = − x a cot ⁡ a x 2 + 2 a 2 ln ⁡ | sin ⁡ a x 2 | + C {displaystyle int {frac {x,dx}{1-cos ax}}=-{frac {x}{a}}cot {frac {ax}{2}}+{frac {2}{a^{2}}}ln left|sin {frac {ax}{2}}right|+C} ∫ cos ⁡ a x d x 1 + cos ⁡ a x = x − 1 a tan ⁡ a x 2 + C {displaystyle int {frac {cos ax,dx}{1+cos ax}}=x-{frac {1}{a}}tan {frac {ax}{2}}+C} ∫ cos ⁡ a x d x 1 − cos ⁡ a x = − x − 1 a cot ⁡ a x 2 + C {displaystyle int {frac {cos ax,dx}{1-cos ax}}=-x-{frac {1}{a}}cot {frac {ax}{2}}+C} ∫ ( cos ⁡ a 1 x ) ( cos ⁡ a 2 x ) d x = sin ⁡ ( ( a 2 − a 1 ) x ) 2 ( a 2 − a 1 ) + sin ⁡ ( ( a 2 + a 1 ) x ) 2 ( a 2 + a 1 ) + C ( | a 1 | ≠ | a 2 | ) {displaystyle int (cos a_{1}x)(cos a_{2}x),dx={frac {sin((a_{2}-a_{1})x)}{2(a_{2}-a_{1})}}+{frac {sin((a_{2}+a_{1})x)}{2(a_{2}+a_{1})}}+Cqquad {mbox{(}}|a_{1}|neq |a_{2}|{mbox{)}}} ∫ tan ⁡ a x d x = − 1 a ln ⁡ | cos ⁡ a x | + C = 1 a ln ⁡ | sec ⁡ a x | + C {displaystyle int tan ax,dx=-{frac {1}{a}}ln |cos ax|+C={frac {1}{a}}ln |sec ax|+C,!} ∫ tan 2 ⁡ x d x = tan ⁡ x − x + C {displaystyle int tan ^{2}{x},dx=tan {x}-x+C} ∫ tan n ⁡ a x d x = 1 a ( n − 1 ) tan n − 1 ⁡ a x − ∫ tan n − 2 ⁡ a x d x ( n ≠ 1 ) {displaystyle int tan ^{n}ax,dx={frac {1}{a(n-1)}}tan ^{n-1}ax-int tan ^{n-2}ax,dxqquad (nneq 1),!} ∫ d x q tan ⁡ a x + p = 1 p 2 + q 2 ( p x + q a ln ⁡ | q sin ⁡ a x + p cos ⁡ a x | ) + C ( p 2 + q 2 ≠ 0 ) {displaystyle int {frac {dx}{qtan ax+p}}={frac {1}{p^{2}+q^{2}}}(px+{frac {q}{a}}ln |qsin ax+pcos ax|)+Cqquad (p^{2}+q^{2}neq 0),!} ∫ d x tan ⁡ a x + 1 = x 2 + 1 2 a ln ⁡ | sin ⁡ a x + cos ⁡ a x | + C {displaystyle int {frac {dx}{tan ax+1}}={frac {x}{2}}+{frac {1}{2a}}ln |sin ax+cos ax|+C,!} ∫ d x tan ⁡ a x − 1 = − x 2 + 1 2 a ln ⁡ | sin ⁡ a x − cos ⁡ a x | + C {displaystyle int {frac {dx}{tan ax-1}}=-{frac {x}{2}}+{frac {1}{2a}}ln |sin ax-cos ax|+C,!} ∫ tan ⁡ a x d x tan ⁡ a x + 1 = x 2 − 1 2 a ln ⁡ | sin ⁡ a x + cos ⁡ a x | + C {displaystyle int {frac {tan ax,dx}{tan ax+1}}={frac {x}{2}}-{frac {1}{2a}}ln |sin ax+cos ax|+C,!} ∫ tan ⁡ a x d x tan ⁡ a x − 1 = x 2 + 1 2 a ln ⁡ | sin ⁡ a x − cos ⁡ a x | + C {displaystyle int {frac {tan ax,dx}{tan ax-1}}={frac {x}{2}}+{frac {1}{2a}}ln |sin ax-cos ax|+C,!} ∫ sec ⁡ a x d x = 1 a ln ⁡ | sec ⁡ a x + tan ⁡ a x | + C {displaystyle int sec {ax},dx={frac {1}{a}}ln {left|sec {ax}+tan {ax}right|}+C} ∫ sec 2 ⁡ x d x = tan ⁡ x + C {displaystyle int sec ^{2}{x},dx=tan {x}+C} ∫ sec n ⁡ a x d x = sec n − 2 ⁡ a x tan ⁡ a x a ( n − 1 ) + n − 2 n − 1 ∫ sec n − 2 ⁡ a x d x ( n ≠ 1 ) {displaystyle int sec ^{n}{ax},dx={frac {sec ^{n-2}{ax}tan {ax}}{a(n-1)}},+,{frac {n-2}{n-1}}int sec ^{n-2}{ax},dxqquad {mbox{(}}nneq 1{mbox{)}},!} ∫ sec n ⁡ x d x = sec n − 2 ⁡ x tan ⁡ x n − 1 + n − 2 n − 1 ∫ sec n − 2 ⁡ x d x {displaystyle int sec ^{n}{x},dx={frac {sec ^{n-2}{x}tan {x}}{n-1}},+,{frac {n-2}{n-1}}int sec ^{n-2}{x},dx} [1] ∫ d x sec ⁡ x + 1 = x − tan ⁡ x 2 + C {displaystyle int {frac {dx}{sec {x}+1}}=x-tan {frac {x}{2}}+C} ∫ d x sec ⁡ x − 1 = − x − cot ⁡ x 2 + C {displaystyle int {frac {dx}{sec {x}-1}}=-x-cot {frac {x}{2}}+C} ∫ csc ⁡ a x d x = − 1 a ln ⁡ | csc ⁡ a x + cot ⁡ a x | + C = 1 a ln ⁡ | csc ⁡ a x − cot ⁡ a x | + C = 1 a ln ⁡ | tan ⁡ ( a x 2 ) | + C {displaystyle {begin{aligned}int csc {ax},dx&=-{frac {1}{a}}ln {left|csc {ax}+cot {ax}right|}+C&={frac {1}{a}}ln {left|csc {ax}-cot {ax}right|}+C&={frac {1}{a}}ln {left|tan {left({frac {ax}{2}}right)}right|}+Cend{aligned}}} ∫ csc 2 ⁡ x d x = − cot ⁡ x + C {displaystyle int csc ^{2}{x},dx=-cot {x}+C} ∫ csc 3 ⁡ x d x = − 1 2 csc ⁡ x cot ⁡ x − 1 2 ln ⁡ | csc ⁡ x + cot ⁡ x | + C = − 1 2 csc ⁡ x cot ⁡ x + 1 2 ln ⁡ | csc ⁡ x − cot ⁡ x | + C {displaystyle {begin{aligned}int csc ^{3}{x},dx&=-{frac {1}{2}}csc xcot x-{frac {1}{2}}ln |csc x+cot x|+C&=-{frac {1}{2}}csc xcot x+{frac {1}{2}}ln |csc x-cot x|+Cend{aligned}}} ∫ csc n ⁡ a x d x = − csc n − 2 ⁡ a x cot ⁡ a x a ( n − 1 ) + n − 2 n − 1 ∫ csc n − 2 ⁡ a x d x ( n ≠ 1 ) {displaystyle int csc ^{n}{ax},dx=-{frac {csc ^{n-2}{ax}cot {ax}}{a(n-1)}},+,{frac {n-2}{n-1}}int csc ^{n-2}{ax},dxqquad {mbox{ (}}nneq 1{mbox{)}}} ∫ d x csc ⁡ x + 1 = x − 2 cot ⁡ x 2 + 1 + C {displaystyle int {frac {dx}{csc {x}+1}}=x-{frac {2}{cot {frac {x}{2}}+1}}+C} ∫ d x csc ⁡ x − 1 = − x + 2 cot ⁡ x 2 − 1 + C {displaystyle int {frac {dx}{csc {x}-1}}=-x+{frac {2}{cot {frac {x}{2}}-1}}+C} ∫ cot ⁡ a x d x = 1 a ln ⁡ | sin ⁡ a x | + C {displaystyle int cot ax,dx={frac {1}{a}}ln |sin ax|+C} ∫ cot 2 ⁡ x d x = − cot ⁡ x − x + C {displaystyle int cot ^{2}{x},dx=-cot {x}-x+C} ∫ cot n ⁡ a x d x = − 1 a ( n − 1 ) cot n − 1 ⁡ a x − ∫ cot n − 2 ⁡ a x d x ( n ≠ 1 ) {displaystyle int cot ^{n}ax,dx=-{frac {1}{a(n-1)}}cot ^{n-1}ax-int cot ^{n-2}ax,dxqquad {mbox{(}}nneq 1{mbox{)}}} ∫ d x 1 + cot ⁡ a x = ∫ tan ⁡ a x d x tan ⁡ a x + 1 = x 2 − 1 2 a ln ⁡ | sin ⁡ a x + cos ⁡ a x | + C {displaystyle int {frac {dx}{1+cot ax}}=int {frac {tan ax,dx}{tan ax+1}}={frac {x}{2}}-{frac {1}{2a}}ln |sin ax+cos ax|+C} ∫ d x 1 − cot ⁡ a x = ∫ tan ⁡ a x d x tan ⁡ a x − 1 = x 2 + 1 2 a ln ⁡ | sin ⁡ a x − cos ⁡ a x | + C {displaystyle int {frac {dx}{1-cot ax}}=int {frac {tan ax,dx}{tan ax-1}}={frac {x}{2}}+{frac {1}{2a}}ln |sin ax-cos ax|+C}

Tích phân một hàm hữu tỉ (phân thức) của sin và cos có thể được tính bằng quy tắc Bioche.

∫ d x cos ⁡ a x ± sin ⁡ a x = 1 a 2 ln ⁡ | tan ⁡ ( a x 2 ± π 8 ) | + C {displaystyle int {frac {dx}{cos axpm sin ax}}={frac {1}{a{sqrt {2}}}}ln left|tan left({frac {ax}{2}}pm {frac {pi }{8}}right)right|+C} ∫ d x ( cos ⁡ a x ± sin ⁡ a x ) 2 = 1 2 a tan ⁡ ( a x ∓ π 4 ) + C {displaystyle int {frac {dx}{(cos axpm sin ax)^{2}}}={frac {1}{2a}}tan left(axmp {frac {pi }{4}}right)+C} ∫ d x ( cos ⁡ x + sin ⁡ x ) n = 1 n − 1 ( sin ⁡ x − cos ⁡ x ( cos ⁡ x + sin ⁡ x ) n − 1 − 2 ( n − 2 ) ∫ d x ( cos ⁡ x + sin ⁡ x ) n − 2 ) {displaystyle int {frac {dx}{(cos x+sin x)^{n}}}={frac {1}{n-1}}left({frac {sin x-cos x}{(cos x+sin x)^{n-1}}}-2(n-2)int {frac {dx}{(cos x+sin x)^{n-2}}}right)} ∫ cos ⁡ a x d x cos ⁡ a x ± sin ⁡ a x = x 2 ± 1 2 a ln ⁡ | sin ⁡ a x ± cos ⁡ a x | + C {displaystyle int {frac {cos ax,dx}{cos axpm sin ax}}={frac {x}{2}}pm {frac {1}{2a}}ln left|sin axpm cos axright|+C} ∫ sin ⁡ a x d x cos ⁡ a x ± sin ⁡ a x = ± x 2 − 1 2 a ln ⁡ | sin ⁡ a x ± cos ⁡ a x | + C {displaystyle int {frac {sin ax,dx}{cos axpm sin ax}}=pm {frac {x}{2}}-{frac {1}{2a}}ln left|sin axpm cos axright|+C} ∫ cos ⁡ a x d x ( sin ⁡ a x ) ( 1 + cos ⁡ a x ) = − 1 4 a tan 2 ⁡ a x 2 + 1 2 a ln ⁡ | tan ⁡ a x 2 | + C {displaystyle int {frac {cos ax,dx}{(sin ax)(1+cos ax)}}=-{frac {1}{4a}}tan ^{2}{frac {ax}{2}}+{frac {1}{2a}}ln left|tan {frac {ax}{2}}right|+C} ∫ cos ⁡ a x d x ( sin ⁡ a x ) ( 1 − cos ⁡ a x ) = − 1 4 a cot 2 ⁡ a x 2 − 1 2 a ln ⁡ | tan ⁡ a x 2 | + C {displaystyle int {frac {cos ax,dx}{(sin ax)(1-cos ax)}}=-{frac {1}{4a}}cot ^{2}{frac {ax}{2}}-{frac {1}{2a}}ln left|tan {frac {ax}{2}}right|+C} ∫ sin ⁡ a x d x ( cos ⁡ a x ) ( 1 + sin ⁡ a x ) = 1 4 a cot 2 ⁡ ( a x 2 + π 4 ) + 1 2 a ln ⁡ | tan ⁡ ( a x 2 + π 4 ) | + C {displaystyle int {frac {sin ax,dx}{(cos ax)(1+sin ax)}}={frac {1}{4a}}cot ^{2}left({frac {ax}{2}}+{frac {pi }{4}}right)+{frac {1}{2a}}ln left|tan left({frac {ax}{2}}+{frac {pi }{4}}right)right|+C} ∫ sin ⁡ a x d x ( cos ⁡ a x ) ( 1 − sin ⁡ a x ) = 1 4 a tan 2 ⁡ ( a x 2 + π 4 ) − 1 2 a ln ⁡ | tan ⁡ ( a x 2 + π 4 ) | + C {displaystyle int {frac {sin ax,dx}{(cos ax)(1-sin ax)}}={frac {1}{4a}}tan ^{2}left({frac {ax}{2}}+{frac {pi }{4}}right)-{frac {1}{2a}}ln left|tan left({frac {ax}{2}}+{frac {pi }{4}}right)right|+C} ∫ ( sin ⁡ a x ) ( cos ⁡ a x ) d x = 1 2 a sin 2 ⁡ a x + C {displaystyle int (sin ax)(cos ax),dx={frac {1}{2a}}sin ^{2}ax+C} ∫ ( sin ⁡ a 1 x ) ( cos ⁡ a 2 x ) d x = − cos ⁡ ( ( a 1 − a 2 ) x ) 2 ( a 1 − a 2 ) − cos ⁡ ( ( a 1 + a 2 ) x ) 2 ( a 1 + a 2 ) + C ( | a 1 | ≠ | a 2 | ) {displaystyle int (sin a_{1}x)(cos a_{2}x),dx=-{frac {cos((a_{1}-a_{2})x)}{2(a_{1}-a_{2})}}-{frac {cos((a_{1}+a_{2})x)}{2(a_{1}+a_{2})}}+Cqquad {mbox{(}}|a_{1}|neq |a_{2}|{mbox{)}}} ∫ ( sin n ⁡ a x ) ( cos ⁡ a x ) d x = 1 a ( n + 1 ) sin n + 1 ⁡ a x + C ( n ≠ − 1 ) {displaystyle int (sin ^{n}ax)(cos ax),dx={frac {1}{a(n+1)}}sin ^{n+1}ax+Cqquad {mbox{(}}nneq -1{mbox{)}}} ∫ ( sin ⁡ a x ) ( cos n ⁡ a x ) d x = − 1 a ( n + 1 ) cos n + 1 ⁡ a x + C ( n ≠ − 1 ) {displaystyle int (sin ax)(cos ^{n}ax),dx=-{frac {1}{a(n+1)}}cos ^{n+1}ax+Cqquad {mbox{(}}nneq -1{mbox{)}}} ∫ ( sin n ⁡ a x ) ( cos m ⁡ a x ) d x = − ( sin n − 1 ⁡ a x ) ( cos m + 1 ⁡ a x ) a ( n + m ) + n − 1 n + m ∫ ( sin n − 2 ⁡ a x ) ( cos m ⁡ a x ) d x ( m , n > 0 ) = ( sin n + 1 ⁡ a x ) ( cos m − 1 ⁡ a x ) a ( n + m ) + m − 1 n + m ∫ ( sin n ⁡ a x ) ( cos m − 2 ⁡ a x ) d x ( m , n > 0 ) {displaystyle {begin{aligned}int (sin ^{n}ax)(cos ^{m}ax),dx&=-{frac {(sin ^{n-1}ax)(cos ^{m+1}ax)}{a(n+m)}}+{frac {n-1}{n+m}}int (sin ^{n-2}ax)(cos ^{m}ax),dxqquad {mbox{(}}m,n>0{mbox{)}}&={frac {(sin ^{n+1}ax)(cos ^{m-1}ax)}{a(n+m)}}+{frac {m-1}{n+m}}int (sin ^{n}ax)(cos ^{m-2}ax),dxqquad {mbox{(}}m,n>0{mbox{)}}end{aligned}}} ∫ d x ( sin ⁡ a x ) ( cos ⁡ a x ) = 1 a ln ⁡ | tan ⁡ a x | + C {displaystyle int {frac {dx}{(sin ax)(cos ax)}}={frac {1}{a}}ln left|tan axright|+C} ∫ d x ( sin ⁡ a x ) ( cos n ⁡ a x ) = 1 a ( n − 1 ) cos n − 1 ⁡ a x + ∫ d x ( sin ⁡ a x ) ( cos n − 2 ⁡ a x ) ( n ≠ 1 ) {displaystyle int {frac {dx}{(sin ax)(cos ^{n}ax)}}={frac {1}{a(n-1)cos ^{n-1}ax}}+int {frac {dx}{(sin ax)(cos ^{n-2}ax)}}qquad {mbox{(}}nneq 1{mbox{)}}} ∫ d x ( sin n ⁡ a x ) ( cos ⁡ a x ) = − 1 a ( n − 1 ) sin n − 1 ⁡ a x + ∫ d x ( sin n − 2 ⁡ a x ) ( cos ⁡ a x ) ( n ≠ 1 ) {displaystyle int {frac {dx}{(sin ^{n}ax)(cos ax)}}=-{frac {1}{a(n-1)sin ^{n-1}ax}}+int {frac {dx}{(sin ^{n-2}ax)(cos ax)}}qquad {mbox{(}}nneq 1{mbox{)}}} ∫ sin ⁡ a x d x cos n ⁡ a x = 1 a ( n − 1 ) cos n − 1 ⁡ a x + C ( n ≠ 1 ) {displaystyle int {frac {sin ax,dx}{cos ^{n}ax}}={frac {1}{a(n-1)cos ^{n-1}ax}}+Cqquad {mbox{(}}nneq 1{mbox{)}}} ∫ sin 2 ⁡ a x d x cos ⁡ a x = − 1 a sin ⁡ a x + 1 a ln ⁡ | tan ⁡ ( π 4 + a x 2 ) | + C {displaystyle int {frac {sin ^{2}ax,dx}{cos ax}}=-{frac {1}{a}}sin ax+{frac {1}{a}}ln left|tan left({frac {pi }{4}}+{frac {ax}{2}}right)right|+C} ∫ sin 2 ⁡ a x d x cos n ⁡ a x = sin ⁡ a x a ( n − 1 ) cos n − 1 ⁡ a x − 1 n − 1 ∫ d x cos n − 2 ⁡ a x ( n ≠ 1 ) {displaystyle int {frac {sin ^{2}ax,dx}{cos ^{n}ax}}={frac {sin ax}{a(n-1)cos ^{n-1}ax}}-{frac {1}{n-1}}int {frac {dx}{cos ^{n-2}ax}}qquad {mbox{(}}nneq 1{mbox{)}}} ∫ sin n ⁡ a x d x cos ⁡ a x = − sin n − 1 ⁡ a x a ( n − 1 ) + ∫ sin n − 2 ⁡ a x d x cos ⁡ a x ( n ≠ 1 ) {displaystyle int {frac {sin ^{n}ax,dx}{cos ax}}=-{frac {sin ^{n-1}ax}{a(n-1)}}+int {frac {sin ^{n-2}ax,dx}{cos ax}}qquad {mbox{(}}nneq 1{mbox{)}}} ∫ sin n ⁡ a x d x cos m ⁡ a x = { sin n + 1 ⁡ a x a ( m − 1 ) cos m − 1 ⁡ a x − n − m + 2 m − 1 ∫ sin n ⁡ a x d x cos m − 2 ⁡ a x ( m ≠ 1 ) sin n − 1 ⁡ a x a ( m − 1 ) cos m − 1 ⁡ a x − n − 1 m − 1 ∫ sin n − 2 ⁡ a x d x cos m − 2 ⁡ a x ( m ≠ 1 ) − sin n − 1 ⁡ a x a ( n − m ) cos m − 1 ⁡ a x + n − 1 n − m ∫ sin n − 2 ⁡ a x d x cos m ⁡ a x ( m ≠ n ) {displaystyle int {frac {sin ^{n}ax,dx}{cos ^{m}ax}}={begin{cases}{dfrac {sin ^{n+1}ax}{a(m-1)cos ^{m-1}ax}}-{dfrac {n-m+2}{m-1}}displaystyle int {dfrac {sin ^{n}ax,dx}{cos ^{m-2}ax}}&{mbox{(}}mneq 1{mbox{)}}{dfrac {sin ^{n-1}ax}{a(m-1)cos ^{m-1}ax}}-{dfrac {n-1}{m-1}}displaystyle int {dfrac {sin ^{n-2}ax,dx}{cos ^{m-2}ax}}&{mbox{(}}mneq 1{mbox{)}}-{dfrac {sin ^{n-1}ax}{a(n-m)cos ^{m-1}ax}}+{dfrac {n-1}{n-m}}displaystyle int {dfrac {sin ^{n-2}ax,dx}{cos ^{m}ax}}&{mbox{(}}mneq n{mbox{)}}end{cases}}} ∫ cos ⁡ a x d x sin n ⁡ a x = − 1 a ( n − 1 ) sin n − 1 ⁡ a x + C ( n ≠ 1 ) {displaystyle int {frac {cos ax,dx}{sin ^{n}ax}}=-{frac {1}{a(n-1)sin ^{n-1}ax}}+Cqquad {mbox{(}}nneq 1{mbox{)}}} ∫ cos 2 ⁡ a x d x sin ⁡ a x = 1 a ( cos ⁡ a x + ln ⁡ | tan ⁡ a x 2 | ) + C {displaystyle int {frac {cos ^{2}ax,dx}{sin ax}}={frac {1}{a}}left(cos ax+ln left|tan {frac {ax}{2}}right|right)+C} ∫ cos 2 ⁡ a x d x sin n ⁡ a x = − 1 n − 1 ( cos ⁡ a x a sin n − 1 ⁡ a x + ∫ d x sin n − 2 ⁡ a x ) ( n ≠ 1 ) {displaystyle int {frac {cos ^{2}ax,dx}{sin ^{n}ax}}=-{frac {1}{n-1}}left({frac {cos ax}{asin ^{n-1}ax}}+int {frac {dx}{sin ^{n-2}ax}}right)qquad {mbox{(}}nneq 1{mbox{)}}} ∫ cos n ⁡ a x d x sin m ⁡ a x = { − cos n + 1 ⁡ a x a ( m − 1 ) sin m − 1 ⁡ a x − n − m + 2 m − 1 ∫ cos n ⁡ a x d x sin m − 2 ⁡ a x ( m ≠ 1 ) − cos n − 1 ⁡ a x a ( m − 1 ) sin m − 1 ⁡ a x − n − 1 m − 1 ∫ cos n − 2 ⁡ a x d x sin m − 2 ⁡ a x ( m ≠ 1 ) cos n − 1 ⁡ a x a ( n − m ) sin m − 1 ⁡ a x + n − 1 n − m ∫ cos n − 2 ⁡ a x d x sin m ⁡ a x ( m ≠ n ) {displaystyle int {frac {cos ^{n}ax,dx}{sin ^{m}ax}}={begin{cases}-{dfrac {cos ^{n+1}ax}{a(m-1)sin ^{m-1}ax}}-{dfrac {n-m+2}{m-1}}displaystyle int {dfrac {cos ^{n}ax,dx}{sin ^{m-2}ax}}&{mbox{(}}mneq 1{mbox{)}}-{dfrac {cos ^{n-1}ax}{a(m-1)sin ^{m-1}ax}}-{dfrac {n-1}{m-1}}displaystyle int {dfrac {cos ^{n-2}ax,dx}{sin ^{m-2}ax}}&{mbox{(}}mneq 1{mbox{)}}{dfrac {cos ^{n-1}ax}{a(n-m)sin ^{m-1}ax}}+{dfrac {n-1}{n-m}}displaystyle int {dfrac {cos ^{n-2}ax,dx}{sin ^{m}ax}}&{mbox{(}}mneq n{mbox{)}}end{cases}}} ∫ sin ⁡ a x tan ⁡ a x d x = 1 a ( ln ⁡ | sec ⁡ a x + tan ⁡ a x | − sin ⁡ a x ) + C {displaystyle int sin axtan ax,dx={frac {1}{a}}(ln |sec ax+tan ax|-sin ax)+C,!} ∫ tan n ⁡ a x d x sin 2 ⁡ a x = 1 a ( n − 1 ) tan n − 1 ⁡ ( a x ) + C ( n ≠ 1 ) {displaystyle int {frac {tan ^{n}ax,dx}{sin ^{2}ax}}={frac {1}{a(n-1)}}tan ^{n-1}(ax)+Cqquad (nneq 1),!} ∫ tan n ⁡ a x d x cos 2 ⁡ a x = 1 a ( n + 1 ) tan n + 1 ⁡ a x + C ( n ≠ − 1 ) {displaystyle int {frac {tan ^{n}ax,dx}{cos ^{2}ax}}={frac {1}{a(n+1)}}tan ^{n+1}ax+Cqquad (nneq -1),!} ∫ cot n ⁡ a x d x sin 2 ⁡ a x = − 1 a ( n + 1 ) cot n + 1 ⁡ a x + C ( n ≠ − 1 ) {displaystyle int {frac {cot ^{n}ax,dx}{sin ^{2}ax}}=-{frac {1}{a(n+1)}}cot ^{n+1}ax+Cqquad (nneq -1),!} ∫ cot n ⁡ a x d x cos 2 ⁡ a x = 1 a ( 1 − n ) tan 1 − n ⁡ a x + C ( n ≠ 1 ) {displaystyle int {frac {cot ^{n}ax,dx}{cos ^{2}ax}}={frac {1}{a(1-n)}}tan ^{1-n}ax+Cqquad (nneq 1),!} ∫ ( sec ⁡ x ) ( tan ⁡ x ) d x = sec ⁡ x + C {displaystyle int (sec x)(tan x),dx=sec x+C} ∫ ( csc ⁡ x ) ( cot ⁡ x ) d x = − csc ⁡ x + C {displaystyle int (csc x)(cot x),dx=-csc x+C} ∫ 0 π 2 sin n ⁡ x d x = ∫ 0 π 2 cos n ⁡ x d x = { n − 1 n ⋅ n − 3 n − 2 ⋯ 3 4 ⋅ 1 2 ⋅ π 2 , n = 2 , 4 , 6 , 8 , … n − 1 n ⋅ n − 3 n − 2 ⋯ 4 5 ⋅ 2 3 , n = 3 , 5 , 7 , 9 , … 1 , n = 1 {displaystyle int _{0}^{frac {pi }{2}}sin ^{n}x,dx=int _{0}^{frac {pi }{2}}cos ^{n}x,dx={begin{cases}{frac {n-1}{n}}cdot {frac {n-3}{n-2}}cdots {frac {3}{4}}cdot {frac {1}{2}}cdot {frac {pi }{2}},&n=2,4,6,8,ldots {frac {n-1}{n}}cdot {frac {n-3}{n-2}}cdots {frac {4}{5}}cdot {frac {2}{3}},&n=3,5,7,9,ldots 1,&n=1end{cases}}} ∫ − c c sin ⁡ x d x = 0 {displaystyle int _{-c}^{c}sin {x},dx=0} ∫ − c c cos ⁡ x d x = 2 ∫ 0 c cos ⁡ x d x = 2 ∫ − c 0 cos ⁡ x d x = 2 sin ⁡ c {displaystyle int _{-c}^{c}cos {x},dx=2int _{0}^{c}cos {x},dx=2int _{-c}^{0}cos {x},dx=2sin {c}} ∫ − c c tan ⁡ x d x = 0 {displaystyle int _{-c}^{c}tan {x},dx=0} ∫ − a 2 a 2 x 2 cos 2 ⁡ n π x a d x = a 3 ( n 2 π 2 − 6 ) 24 n 2 π 2 {displaystyle int _{-{frac {a}{2}}}^{frac {a}{2}}x^{2}cos ^{2}{frac {npi x}{a}},dx={frac {a^{3}(n^{2}pi ^{2}-6)}{24n^{2}pi ^{2}}}qquad } (n là số nguyên dương lẻ) ∫ − a 2 a 2 x 2 sin 2 ⁡ n π x a d x = a 3 ( n 2 π 2 − 6 ( − 1 ) n ) 24 n 2 π 2 = a 3 24 ( 1 − 6 ( − 1 ) n n 2 π 2 ) {displaystyle int _{frac {-a}{2}}^{frac {a}{2}}x^{2}sin ^{2}{frac {npi x}{a}},dx={frac {a^{3}(n^{2}pi ^{2}-6(-1)^{n})}{24n^{2}pi ^{2}}}={frac {a^{3}}{24}}(1-6{frac {(-1)^{n}}{n^{2}pi ^{2}}})qquad } (n là số nguyên dương) ∫ 0 2 π sin 2 m + 1 ⁡ x cos 2 n + 1 ⁡ x d x = 0 m , n ∈ Z {displaystyle int _{0}^{2pi }sin ^{2m+1}{x}cos ^{2n+1}{x},dx=0qquad m,nin mathbb {Z} }

• Gradshteĭn, I. S. (2015). Table of Integrals, Series, and Products. Waltham, MA: Academic Press. ISBN 978-0-12-384933-5. OCLC 893676501.