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integral of 2/((25+x^2)^{3/2)}

解答

\int \frac{2}{( 25 + x ^{2} ) ^{\frac{3}{2}}} d x

解答

\frac{2 x}{25 ( 25 + x ^{2} ) ^{\frac{1}{2}}} + C

求解步骤

\int \frac{2}{( 25 + x ^{2} ) ^{\frac{3}{2}}} d x

提出常数:

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

= 2 \cdot \int \frac{1}{( 25 + x ^{2} ) ^{\frac{3}{2}}} d x

使用三角换元法

= 2 \cdot \int \frac{1}{25 sec ( u )} d u

提出常数:

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

= 2 \cdot \frac{1}{25} \cdot \int \frac{1}{sec ( u )} d u

使用三角恒等式改写

= 2 \cdot \frac{1}{25} \cdot \int cos (u) d u

使用积分常见定则:

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

= 2 \cdot \frac{1}{25} sin (u)

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

代回

= 2 \cdot \frac{1}{25} sin (arctan (\frac{1}{5} x))

化简

2 \cdot \frac{1}{25} sin (arctan (\frac{1}{5} x)) : \frac{2 x}{25 ( 25 + x ^{2} ) ^{\frac{1}{2}}}

= \frac{2 x}{25 ( 25 + x ^{2} ) ^{\frac{1}{2}}}

解答补常数

= \frac{2 x}{25 ( 25 + x ^{2} ) ^{\frac{1}{2}}} + C

作图

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