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受欢迎的微积分 >

integral of sin(7x)sec^5(7x)

解答

\int sin (7 x) sec^{5} (7 x) d x

解答

\frac{1}{28} sec^{4} (7 x) + C

求解步骤

\int sin (7 x) sec^{5} (7 x) d x

使用三角恒等式改写

= \int \frac{tan ( 7 x )}{cos ^{4} ( 7 x )} d x

使用基本三角恒等式:

\frac{1}{cos ( x )} = sec (x)

= \int sec^{4} (7 x) tan (7 x) d x

使用换元积分法

= \int sec^{4} (u) tan (u) \frac{1}{7} d u

提出常数:

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

= \frac{1}{7} \cdot \int sec^{4} (u) tan (u) d u

使用换元积分法

= \frac{1}{7} \cdot \int v^{3} d v

使用幂法则

= \frac{1}{7} \cdot \frac{v ^{4}}{4}

代回

= \frac{1}{7} \cdot \frac{sec ^{4} ( 7 x )}{4}

化简

\frac{1}{7} \cdot \frac{sec ^{4} ( 7 x )}{4} : \frac{1}{28} sec^{4} (7 x)

= \frac{1}{28} sec^{4} (7 x)

解答补常数

= \frac{1}{28} sec^{4} (7 x) + C

作图

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