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受欢迎的微积分 >

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

解答

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

解答

2 (\frac{1}{5} (x^{2} + 4)^{\frac{5}{2}} - \frac{4}{3} (x^{2} + 4)^{\frac{3}{2}}) + C

求解步骤

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

提出常数:

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

= 2 \cdot \int x^{3} x^{2} + 4 d x

使用换元积分法

= 2 \cdot \int (u^{2} - 4) u^{2} d u

乘开

(u^{2} - 4) u^{2} : u^{4} - 4 u^{2}

= 2 \cdot \int u^{4} - 4 u^{2} 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 u^{4} d u - \int 4 u^{2} d u)

\int u^{4} d u = \frac{u ^{5}}{5}

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

= 2 (\frac{u ^{5}}{5} - \frac{4 u ^{3}}{3})

u = x^{2} + 4

代回

= 2 (\frac{( x ^{2} + 4 ) ^{5}}{5} - \frac{4 ( x ^{2} + 4 ) ^{3}}{3})

化简

2 (\frac{( x ^{2} + 4 ) ^{5}}{5} - \frac{4 ( x ^{2} + 4 ) ^{3}}{3}) : 2 (\frac{1}{5} (x^{2} + 4)^{\frac{5}{2}} - \frac{4}{3} (x^{2} + 4)^{\frac{3}{2}})

= 2 (\frac{1}{5} (x^{2} + 4)^{\frac{5}{2}} - \frac{4}{3} (x^{2} + 4)^{\frac{3}{2}})

解答补常数

= 2 (\frac{1}{5} (x^{2} + 4)^{\frac{5}{2}} - \frac{4}{3} (x^{2} + 4)^{\frac{3}{2}}) + C

作图

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