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受欢迎的微积分 >

integral of (x^2)/(4+x^2)

解答

\int \frac{x ^{2}}{4 + x ^{2}} d x

解答

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

求解步骤

\int \frac{x ^{2}}{4 + x ^{2}} d x

\frac{x ^{2}}{4 + x ^{2}} = - \frac{4}{4 + x ^{2}} + 1

= \int - \frac{4}{4 + x ^{2}} + 1 d x

使用换元积分法

= \int 2 (- \frac{1}{1 + u ^{2}} + 1) d u

提出常数:

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

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

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

\int 1 d u = u

= 2 (- arctan (u) + u)

u = \frac{x}{2}

代回

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

化简

2 (- arctan (\frac{x}{2}) + \frac{x}{2}) : - 2 arctan (\frac{x}{2}) + x

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

解答补常数

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

作图

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