integral of x/((7+5x^2)^{3/2)}

解答

\int \frac{x}{( 7 + 5 x ^{2} ) ^{\frac{3}{2}}} d x

- \frac{1}{5 ( 5 x ^{2} + 7 ) ^{\frac{1}{2}}} + C

求解步骤

\int \frac{x}{( 7 + 5 x ^{2} ) ^{\frac{3}{2}}} d x

使用换元积分法

= \int \frac{1}{10 u ^{\frac{3}{2}}} d u

提出常数:

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

= \frac{1}{10} \cdot \int \frac{1}{u ^{\frac{3}{2}}} d u

使用幂法则

= \frac{1}{10} (- \frac{2}{u ^{\frac{1}{2}}})

u = 7 + 5 x^{2}

代回

= \frac{1}{10} (- \frac{2}{( 7 + 5 x ^{2} ) ^{\frac{1}{2}}})

化简

\frac{1}{10} (- \frac{2}{( 7 + 5 x ^{2} ) ^{\frac{1}{2}}}) : - \frac{1}{5 ( 5 x ^{2} + 7 ) ^{\frac{1}{2}}}

= - \frac{1}{5 ( 5 x ^{2} + 7 ) ^{\frac{1}{2}}}

解答补常数

= - \frac{1}{5 ( 5 x ^{2} + 7 ) ^{\frac{1}{2}}} + C

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