integral of x(9x^2+5)^3

解答

\int x (9 x^{2} + 5)^{3} d x

\frac{1}{72} (9 x^{2} + 5)^{4} + C

求解步骤

\int x (9 x^{2} + 5)^{3} d x

使用换元积分法

= \int \frac{u ^{3}}{18} d u

提出常数:

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

= \frac{1}{18} \cdot \int u^{3} d u

使用幂法则

= \frac{1}{18} \cdot \frac{u ^{4}}{4}

u = 9 x^{2} + 5

代回

= \frac{1}{18} \cdot \frac{( 9 x ^{2} + 5 ) ^{4}}{4}

化简

\frac{1}{18} \cdot \frac{( 9 x ^{2} + 5 ) ^{4}}{4} : \frac{1}{72} (9 x^{2} + 5)^{4}

= \frac{1}{72} (9 x^{2} + 5)^{4}

解答补常数

= \frac{1}{72} (9 x^{2} + 5)^{4} + C

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