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受欢迎的微积分 >

integral of 1/((5+x)sqrt(1+x))

解答

\int \frac{1}{( 5 + x ) 1 + x} d x

解答

arctan (\frac{1}{2} x + 1) + C

求解步骤

\int \frac{1}{( 5 + x ) 1 + x} d x

使用换元积分法

= \int \frac{1}{u u - 4} d u

使用换元积分法

= \int \frac{2}{v ^{2} + 4} d v

提出常数:

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

= 2 \cdot \int \frac{1}{v ^{2} + 4} d v

使用换元积分法

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

提出常数:

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

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

使用积分常见定则:

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

= 2 \cdot \frac{1}{2} arctan (w)

代回

= 2 \cdot \frac{1}{2} arctan (\frac{5 + x - 4}{2})

化简

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

= arctan (\frac{1}{2} x + 1)

解答补常数

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

作图

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