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受欢迎的微积分 >

integral of sin^2(10t)sec^4(10t)

解答

\int sin^{2} (10 t) sec^{4} (10 t) d t

解答

\frac{1}{30} tan^{3} (10 t) + C

求解步骤

\int sin^{2} (10 t) sec^{4} (10 t) d t

使用三角恒等式改写

= \int \frac{sin ^{2} ( 10 t )}{cos ^{4} ( 10 t )} d t

使用换元积分法

= \int \frac{sin ^{2} ( u )}{10 cos ^{4} ( u )} d u

提出常数:

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

= \frac{1}{10} \cdot \int \frac{sin ^{2} ( u )}{cos ^{4} ( u )} d u

使用三角恒等式改写

= \frac{1}{10} \cdot \int \frac{tan ^{2} ( u )}{cos ^{2} ( u )} d u

使用换元积分法

= \frac{1}{10} \cdot \int v^{2} d v

使用幂法则

= \frac{1}{10} \cdot \frac{v ^{3}}{3}

代回

= \frac{1}{10} \cdot \frac{tan ^{3} ( 10 t )}{3}

化简

\frac{1}{10} \cdot \frac{tan ^{3} ( 10 t )}{3} : \frac{1}{30} tan^{3} (10 t)

= \frac{1}{30} tan^{3} (10 t)

解答补常数

= \frac{1}{30} tan^{3} (10 t) + C

作图

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