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受欢迎的微积分 >

integral of sec^2(3x)tan^3(3x)

解答

\int sec^{2} (3 x) tan^{3} (3 x) d x

解答

\frac{1}{12} tan^{4} (3 x) + C

求解步骤

\int sec^{2} (3 x) tan^{3} (3 x) d x

使用换元积分法

= \int sec^{2} (u) tan^{3} (u) \frac{1}{3} d u

提出常数:

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

= \frac{1}{3} \cdot \int sec^{2} (u) tan^{3} (u) d u

使用三角恒等式改写

= \frac{1}{3} \cdot \int (1 + tan^{2} (u)) tan^{3} (u) d u

使用换元积分法

= \frac{1}{3} \cdot \int v^{3} d v

使用幂法则

= \frac{1}{3} \cdot \frac{v ^{4}}{4}

代回

= \frac{1}{3} \cdot \frac{tan ^{4} ( 3 x )}{4}

化简

\frac{1}{3} \cdot \frac{tan ^{4} ( 3 x )}{4} : \frac{1}{12} tan^{4} (3 x)

= \frac{1}{12} tan^{4} (3 x)

解答补常数

= \frac{1}{12} tan^{4} (3 x) + C

作图

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