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受欢迎的微积分 >

integral of (cos(5x))/(sin^2(5x))

解答

\int \frac{cos ( 5 x )}{sin ^{2} ( 5 x )} d x

解答

- \frac{1}{5} csc (5 x) + C

求解步骤

\int \frac{cos ( 5 x )}{sin ^{2} ( 5 x )} d x

使用基本三角恒等式:

\frac{cos ( x )}{sin ( x )} = cot (x)

= \int \frac{cot ( 5 x )}{sin ( 5 x )} d x

使用换元积分法

= \int \frac{cot ( u )}{5 sin ( u )} d u

提出常数:

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

= \frac{1}{5} \cdot \int \frac{cot ( u )}{sin ( u )} d u

使用三角恒等式改写

= \frac{1}{5} \cdot \int \frac{cos ( u )}{sin ^{2} ( u )} d u

使用换元积分法

= \frac{1}{5} \cdot \int \frac{1}{v ^{2}} d v

使用幂法则

= \frac{1}{5} (- \frac{1}{v})

代回

= \frac{1}{5} (- \frac{1}{sin ( 5 x )})

化简

\frac{1}{5} (- \frac{1}{sin ( 5 x )}) : - \frac{1}{5} csc (5 x)

= - \frac{1}{5} csc (5 x)

解答补常数

= - \frac{1}{5} csc (5 x) + C

作图

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