integral of 3x(2x+5)^2

解答

\int 3 x (2 x + 5)^{2} d x

3 x^{4} + 20 x^{3} + \frac{75 x ^{2}}{2} + C

求解步骤

\int 3 x (2 x + 5)^{2} d x

提出常数:

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

= 3 \cdot \int x (2 x + 5)^{2} d x

乘开

x (2 x + 5)^{2} : 4 x^{3} + 20 x^{2} + 25 x

= 3 \cdot \int 4 x^{3} + 20 x^{2} + 25 x d x

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

= 3 (\int 4 x^{3} d x + \int 20 x^{2} d x + \int 25 x d x)

\int 4 x^{3} d x = x^{4}

\int 20 x^{2} d x = \frac{20 x ^{3}}{3}

\int 25 x d x = \frac{25 x ^{2}}{2}

= 3 (x^{4} + \frac{20 x ^{3}}{3} + \frac{25 x ^{2}}{2})

化简

3 (x^{4} + \frac{20 x ^{3}}{3} + \frac{25 x ^{2}}{2}) : 3 x^{4} + 20 x^{3} + \frac{75 x ^{2}}{2}

= 3 x^{4} + 20 x^{3} + \frac{75 x ^{2}}{2}

解答补常数

= 3 x^{4} + 20 x^{3} + \frac{75 x ^{2}}{2} + C

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