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受欢迎的微积分 >

integral of sec^2(5x)tan^4(5x)

解答

\int sec^{2} (5 x) tan^{4} (5 x) d x

解答

\frac{1}{25} tan^{5} (5 x) + C

求解步骤

\int sec^{2} (5 x) tan^{4} (5 x) d x

使用换元积分法

= \int sec^{2} (u) tan^{4} (u) \frac{1}{5} 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 sec^{2} (u) tan^{4} (u) d u

使用三角恒等式改写

= \frac{1}{5} \cdot \int (1 + tan^{2} (u)) tan^{4} (u) d u

使用换元积分法

= \frac{1}{5} \cdot \int v^{4} d v

使用幂法则

= \frac{1}{5} \cdot \frac{v ^{5}}{5}

代回

= \frac{1}{5} \cdot \frac{tan ^{5} ( 5 x )}{5}

化简

\frac{1}{5} \cdot \frac{tan ^{5} ( 5 x )}{5} : \frac{1}{25} tan^{5} (5 x)

= \frac{1}{25} tan^{5} (5 x)

解答补常数

= \frac{1}{25} tan^{5} (5 x) + C

作图

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