integral of (cos(2x))/((1+sin(2x))^4)

解答

\int \frac{cos ( 2 x )}{( 1 + sin ( 2 x ) ) ^{4}} d x

- \frac{1}{6 ( 1 + sin ( 2 x ) ) ^{3}} + C

求解步骤

\int \frac{cos ( 2 x )}{( 1 + sin ( 2 x ) ) ^{4}} d x

使用换元积分法

= \int \frac{1}{2 u ^{4}} d u

提出常数:

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

= \frac{1}{2} \cdot \int \frac{1}{u ^{4}} d u

使用幂法则

= \frac{1}{2} (- \frac{1}{3 u ^{3}})

u = 1 + sin (2 x)

代回

= \frac{1}{2} (- \frac{1}{3 ( 1 + sin ( 2 x ) ) ^{3}})

化简

\frac{1}{2} (- \frac{1}{3 ( 1 + sin ( 2 x ) ) ^{3}}) : - \frac{1}{6 ( 1 + sin ( 2 x ) ) ^{3}}

= - \frac{1}{6 ( 1 + sin ( 2 x ) ) ^{3}}

解答补常数

= - \frac{1}{6 ( 1 + sin ( 2 x ) ) ^{3}} + C

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