integral of tan(7x)

解答

\int tan (7 x) d x

- \frac{1}{7} ln ∣ cos (7 x) ∣ + C

求解步骤

\int tan (7 x) d x

使用三角恒等式改写

= \int \frac{sin ( 7 x )}{cos ( 7 x )} d x

使用换元积分法

= \int - \frac{1}{7 u} 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 \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - \frac{1}{7} ln ∣ u ∣

u = cos (7 x)

代回

= - \frac{1}{7} ln ∣ cos (7 x) ∣

解答补常数

= - \frac{1}{7} ln ∣ cos (7 x) ∣ + C

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