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受欢迎的微积分 >

integral of 2/((x^2+1)^2)

解答

\int \frac{2}{( x ^{2} + 1 ) ^{2}} d x

解答

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

求解步骤

\int \frac{2}{( x ^{2} + 1 ) ^{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}{( x ^{2} + 1 ) ^{2}} d x

使用三角换元法

= 2 \cdot \int \frac{1}{sec ^{4} ( u ) cos ^{2} ( u )} d u

使用三角恒等式改写

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

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 1 + cos (2 u) d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

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

\int 1 d u = u

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

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

u = arctan (x)

代回

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

化简

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

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

解答补常数

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

作图

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