integral of (3t^2)/(sqrt(t^3+5))

解答

\int \frac{3 t ^{2}}{t ^{3} + 5} d t

2 t^{3} + 5 + C

求解步骤

\int \frac{3 t ^{2}}{t ^{3} + 5} d t

提出常数:

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

= 3 \cdot \int \frac{t ^{2}}{t ^{3} + 5} d t

使用换元积分法

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

常数积分:

\int a d x = a x

= 3 \cdot \frac{2}{3} u

u = t^{3} + 5

代回

= 3 \cdot \frac{2}{3} t^{3} + 5

化简

3 \cdot \frac{2}{3} t^{3} + 5 : 2 t^{3} + 5

= 2 t^{3} + 5

解答补常数

= 2 t^{3} + 5 + C

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