integral of (3x^2)/(b^{5x^3)}

解答

\int \frac{3 x ^{2}}{b ^{5 x^{3}}} d x

- \frac{1}{5 b ^{5 x^{3}} ln ( b )} + C

求解步骤

\int \frac{3 x ^{2}}{b ^{5 x^{3}}} d x

提出常数:

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

= 3 \cdot \int \frac{x ^{2}}{b ^{5 x^{3}}} d x

使用换元积分法

: \int \frac{1}{15 u ^{2} ln ( b )} d u

= 3 \cdot \int \frac{1}{15 u ^{2} ln ( b )} d u

提出常数:

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

= 3 \cdot \frac{1}{15 ln ( b )} \cdot \int \frac{1}{u ^{2}} d u

使用幂法则

: - \frac{1}{u}

= 3 \cdot \frac{1}{15 ln ( b )} (- \frac{1}{u})

u = b^{5 x^{3}}

代回

= 3 \cdot \frac{1}{15 ln ( b )} (- \frac{1}{b ^{5 x^{3}}})

化简

3 \cdot \frac{1}{15 ln ( b )} (- \frac{1}{b ^{5 x^{3}}}) : - \frac{1}{5 b ^{5 x^{3}} ln ( b )}

= - \frac{1}{5 b ^{5 x^{3}} ln ( b )}

解答补常数

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

\int \frac{2 x + 3}{x ^{2} + 3 x} d x

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\int sin (x) cos^{8} (x) d x

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\int (x - 7) d x