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受欢迎的 >

integral of 2/(sqrt(x)(x+5))

解答

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

解答

\frac{4}{5} arctan (\frac{x}{5}) + C

求解步骤

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

使用换元积分法

: \int \frac{2}{u ^{2} + 5} d u

= 2 \cdot \int \frac{2}{u ^{2} + 5} d u

提出常数:

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

= 2 \cdot 2 \cdot \int \frac{1}{u ^{2} + 5} d u

使用换元积分法

= 2 \cdot 2 \cdot \int \frac{1}{5 ( v ^{2} + 1 )} d v

提出常数:

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

= 2 \cdot 2 \cdot \frac{1}{5} \cdot \int \frac{1}{v ^{2} + 1} d v

使用积分常见定则:

\int \frac{1}{v ^{2} + 1} d v = arctan (v)

= 2 \cdot 2 \cdot \frac{1}{5} arctan (v)

代回

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

化简

2 \cdot 2 \cdot \frac{1}{5} arctan (\frac{x}{5}) : \frac{4}{5} arctan (\frac{x}{5})

= \frac{4}{5} arctan (\frac{x}{5})

解答补常数

= \frac{4}{5} arctan (\frac{x}{5}) + C

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