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受欢迎的微积分 >

integral of (x^4)/(3+x^2)

解答

\int \frac{x ^{4}}{3 + x ^{2}} d x

解答

\frac{x ^{3}}{3} - 3 x + 33 arctan (\frac{x}{3}) + C

求解步骤

\int \frac{x ^{4}}{3 + x ^{2}} d x

使用换元积分法

= \int \frac{3 3 u ^{4}}{1 + u ^{2}} d u

提出常数:

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

= 33 \cdot \int \frac{u ^{4}}{1 + u ^{2}} d u

长除

\frac{u ^{4}}{1 + u ^{2}} : u^{2} - 1 + \frac{1}{u ^{2} + 1}

= 33 \cdot \int u^{2} - 1 + \frac{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

= 33 (\int u^{2} d u - \int 1 d u + \int \frac{1}{u ^{2} + 1} d u)

\int u^{2} d u = \frac{u ^{3}}{3}

\int 1 d u = u

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

= 33 (\frac{u ^{3}}{3} - u + arctan (u))

u = \frac{x}{3}

代回

= 33 \frac{( \frac{x}{3} ) ^{3}}{3} - \frac{x}{3} + arctan (\frac{x}{3})

化简

33 \frac{( \frac{x}{3} ) ^{3}}{3} - \frac{x}{3} + arctan (\frac{x}{3}) : \frac{x ^{3}}{3} - 3 x + 33 arctan (\frac{x}{3})

= \frac{x ^{3}}{3} - 3 x + 33 arctan (\frac{x}{3})

解答补常数

= \frac{x ^{3}}{3} - 3 x + 33 arctan (\frac{x}{3}) + C

作图

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