integral of x^3sqrt(1+(3x^2)^2)

解答

\int x^{3} 1 + (3 x^{2})^{2} d x

\frac{1}{54} (1 + 9 x^{4})^{\frac{3}{2}} + C

求解步骤

\int x^{3} 1 + (3 x^{2})^{2} d x

使用换元积分法

= \int \frac{u}{36} d u

提出常数:

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

= \frac{1}{36} \cdot \int u d u

使用幂法则

= \frac{1}{36} \cdot \frac{2}{3} u^{\frac{3}{2}}

u = 1 + (3 x^{2})^{2}

代回

= \frac{1}{36} \cdot \frac{2}{3} (1 + (3 x^{2})^{2})^{\frac{3}{2}}

化简

\frac{1}{36} \cdot \frac{2}{3} (1 + (3 x^{2})^{2})^{\frac{3}{2}} : \frac{1}{54} (1 + 9 x^{4})^{\frac{3}{2}}

= \frac{1}{54} (1 + 9 x^{4})^{\frac{3}{2}}

解答补常数

= \frac{1}{54} (1 + 9 x^{4})^{\frac{3}{2}} + C

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