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受欢迎的微积分 >

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

解答

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

解答

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

求解步骤

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

提出常数:

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

= - 5 \cdot \int \frac{1}{( x ^{2} + 5 ) ^{\frac{3}{2}}} d x

使用三角换元法

= - 5 \cdot \int \frac{1}{5 sec ( u )} d u

提出常数:

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

= - 5 \cdot \frac{1}{5} \cdot \int \frac{1}{sec ( u )} d u

使用三角恒等式改写

= - 5 \cdot \frac{1}{5} \cdot \int cos (u) d u

使用积分常见定则:

\int cos (u) d u = sin (u)

= - 5 \cdot \frac{1}{5} sin (u)

u = arctan (\frac{1}{5 ^{\frac{1}{2}}} x)

代回

= - 5 \cdot \frac{1}{5} sin (arctan (\frac{1}{5 ^{\frac{1}{2}}} x))

化简

- 5 \cdot \frac{1}{5} sin (arctan (\frac{1}{5 ^{\frac{1}{2}}} x)) : - \frac{x}{( 5 + x ^{2} ) ^{\frac{1}{2}}}

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

解答补常数

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

作图

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