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受欢迎的微积分 >

integral of-4cot(6x^2)2x

解答

\int - 4 cot (6 x^{2}) 2 x d x

解答

- \frac{2}{3} ln sin (6 x^{2}) + C

求解步骤

\int - 4 cot (6 x^{2}) \cdot 2 x d x

提出常数:

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

= - 4 \cdot 2 \cdot \int cot (6 x^{2}) x d x

化简

= - 8 \cdot \int cot (6 x^{2}) x d x

使用换元积分法

= - 8 \cdot \int \frac{cot ( u )}{12} d u

提出常数:

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

= - 8 \cdot \frac{1}{12} \cdot \int cot (u) d u

使用三角恒等式改写

= - 8 \cdot \frac{1}{12} \cdot \int \frac{cos ( u )}{sin ( u )} d u

使用换元积分法

= - 8 \cdot \frac{1}{12} \cdot \int \frac{1}{v} d v

使用积分常见定则:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - 8 \cdot \frac{1}{12} ln ∣ v ∣

代回

= - 8 \cdot \frac{1}{12} ln sin (6 x^{2})

化简

- 8 \cdot \frac{1}{12} ln sin (6 x^{2}) : - \frac{2}{3} ln sin (6 x^{2})

= - \frac{2}{3} ln sin (6 x^{2})

解答补常数

= - \frac{2}{3} ln sin (6 x^{2}) + C

作图

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