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受欢迎的微积分 >

integral of 1/3 tan(3x)

解答

\int \frac{1}{3} tan (3 x) d x

解答

- \frac{1}{9} ln ∣ cos (3 x) ∣ + C

求解步骤

\int \frac{1}{3} tan (3 x) d x

提出常数:

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

= \frac{1}{3} \cdot \int tan (3 x) d x

使用三角恒等式改写

= \frac{1}{3} \cdot \int \frac{sin ( 3 x )}{cos ( 3 x )} d x

使用换元积分法

= \frac{1}{3} \cdot \int - \frac{1}{3 u} d u

提出常数:

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

= \frac{1}{3} (- \frac{1}{3} \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}{3} (- \frac{1}{3} ln ∣ u ∣)

u = cos (3 x)

代回

= \frac{1}{3} (- \frac{1}{3} ln ∣ cos (3 x) ∣)

化简

\frac{1}{3} (- \frac{1}{3} ln ∣ cos (3 x) ∣) : - \frac{1}{9} ln ∣ cos (3 x) ∣

= - \frac{1}{9} ln ∣ cos (3 x) ∣

解答补常数

= - \frac{1}{9} ln ∣ cos (3 x) ∣ + C

作图

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