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受欢迎的微积分 >

integral of 8(tan(8x))

解答

\int 8 (tan (8 x)) d x

解答

- ln ∣ cos (8 x) ∣ + C

求解步骤

\int 8 tan (8 x) d x

提出常数:

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

= 8 \cdot \int tan (8 x) d x

使用三角恒等式改写

= 8 \cdot \int \frac{sin ( 8 x )}{cos ( 8 x )} d x

使用换元积分法

= 8 \cdot \int - \frac{1}{8 u} d u

提出常数:

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

= 8 (- \frac{1}{8} \cdot \int \frac{1}{u} d u)

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 8 (- \frac{1}{8} ln ∣ u ∣)

u = cos (8 x)

代回

= 8 (- \frac{1}{8} ln ∣ cos (8 x) ∣)

化简

8 (- \frac{1}{8} ln ∣ cos (8 x) ∣) : - ln ∣ cos (8 x) ∣

= - ln ∣ cos (8 x) ∣

解答补常数

= - ln ∣ cos (8 x) ∣ + C

作图

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