integral of (sin(t))/((8+cos(t))^6)

解答

\int \frac{sin ( t )}{( 8 + cos ( t ) ) ^{6}} d t

- \frac{cos ( t )}{( 8 + cos ( t ) ) ^{6}} + \frac{6}{5 ( 8 + cos ( t ) ) ^{5}} - \frac{8}{( 8 + cos ( t ) ) ^{6}} + C

求解步骤

\int \frac{sin ( t )}{( 8 + cos ( t ) ) ^{6}} d t

使用换元积分法

= \int - \frac{1}{( 8 + u ) ^{6}} d u

提出常数:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int \frac{1}{( 8 + u ) ^{6}} d u

使用分布积分法

= - (\frac{u}{( 8 + u ) ^{6}} - \int - \frac{6 u}{( 8 + u ) ^{7}} d u)

\int - \frac{6 u}{( 8 + u ) ^{7}} d u = - 6 (- \frac{1}{5 ( 8 + u ) ^{5}} + \frac{4}{3 ( 8 + u ) ^{6}})

= - (\frac{u}{( 8 + u ) ^{6}} - (- 6 (- \frac{1}{5 ( 8 + u ) ^{5}} + \frac{4}{3 ( 8 + u ) ^{6}})))

u = cos (t)

代回

= - (\frac{cos ( t )}{( 8 + cos ( t ) ) ^{6}} - (- 6 (- \frac{1}{5 ( 8 + cos ( t ) ) ^{5}} + \frac{4}{3 ( 8 + cos ( t ) ) ^{6}})))

化简

- (\frac{cos ( t )}{( 8 + cos ( t ) ) ^{6}} - (- 6 (- \frac{1}{5 ( 8 + cos ( t ) ) ^{5}} + \frac{4}{3 ( 8 + cos ( t ) ) ^{6}}))) : - \frac{cos ( t )}{( 8 + cos ( t ) ) ^{6}} + \frac{6}{5 ( 8 + cos ( t ) ) ^{5}} - \frac{8}{( 8 + cos ( t ) ) ^{6}}

= - \frac{cos ( t )}{( 8 + cos ( t ) ) ^{6}} + \frac{6}{5 ( 8 + cos ( t ) ) ^{5}} - \frac{8}{( 8 + cos ( t ) ) ^{6}}

解答补常数

= - \frac{cos ( t )}{( 8 + cos ( t ) ) ^{6}} + \frac{6}{5 ( 8 + cos ( t ) ) ^{5}} - \frac{8}{( 8 + cos ( t ) ) ^{6}} + C

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