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受欢迎的微积分 >

integral of (4x^2)/(sin(x^3-5))

解答

\int \frac{4 x ^{2}}{sin ( x ^{3} - 5 )} d x

解答

\frac{4}{3} ln tan (\frac{1}{2} (x^{3} - 5)) + C

求解步骤

\int \frac{4 x ^{2}}{sin ( x ^{3} - 5 )} d x

提出常数:

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

= 4 \cdot \int \frac{x ^{2}}{sin ( x ^{3} - 5 )} d x

使用换元积分法

= 4 \cdot \int \frac{1}{3 sin ( u )} d u

提出常数:

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

= 4 \cdot \frac{1}{3} \cdot \int \frac{1}{sin ( u )} d u

使用换元积分法

= 4 \cdot \frac{1}{3} \cdot \int \frac{1}{v} d v

使用积分常见定则:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 4 \cdot \frac{1}{3} ln ∣ v ∣

代回

= 4 \cdot \frac{1}{3} ln tan (\frac{x ^{3} - 5}{2})

化简

4 \cdot \frac{1}{3} ln tan (\frac{x ^{3} - 5}{2}) : \frac{4}{3} ln tan (\frac{1}{2} (x^{3} - 5))

= \frac{4}{3} ln tan (\frac{1}{2} (x^{3} - 5))

解答补常数

= \frac{4}{3} ln tan (\frac{1}{2} (x^{3} - 5)) + C

作图

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