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受欢迎的微积分 >

integral of 2cot(2x)

解答

\int 2 cot (2 x) d x

解答

ln ∣ sin (2 x) ∣ + C

求解步骤

\int 2 cot (2 x) d x

提出常数:

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

= 2 \cdot \int cot (2 x) d x

使用三角恒等式改写

= 2 \cdot \int \frac{cos ( 2 x )}{sin ( 2 x )} d x

使用换元积分法

= 2 \cdot \int \frac{1}{2 u} d u

提出常数:

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

= 2 \cdot \frac{1}{2} \cdot \int \frac{1}{u} d u

使用积分常见定则:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 2 \cdot \frac{1}{2} ln ∣ u ∣

u = sin (2 x)

代回

= 2 \cdot \frac{1}{2} ln ∣ sin (2 x) ∣

化简

2 \cdot \frac{1}{2} ln ∣ sin (2 x) ∣ : ln ∣ sin (2 x) ∣

= ln ∣ sin (2 x) ∣

解答补常数

= ln ∣ sin (2 x) ∣ + C

作图

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