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受欢迎的微积分 >

integral of (15x\sqrt[4]{3x^2+4})/4

解答

\int \frac{15 x 4 3 x ^{2} + 4}{4} d x

解答

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

求解步骤

\int \frac{15 x 4 3 x ^{2} + 4}{4} d x

提出常数:

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

= \frac{15}{4} \cdot \int x 4 3 x^{2} + 4 d x

使用换元积分法

= \frac{15}{4} \cdot \int \frac{4 u}{6} d u

提出常数:

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

= \frac{15}{4} \cdot \frac{1}{6} \cdot \int 4 u d u

使用幂法则

= \frac{15}{4} \cdot \frac{1}{6} \cdot \frac{4}{5} u^{\frac{5}{4}}

u = 3 x^{2} + 4

代回

= \frac{15}{4} \cdot \frac{1}{6} \cdot \frac{4}{5} (3 x^{2} + 4)^{\frac{5}{4}}

化简

\frac{15}{4} \cdot \frac{1}{6} \cdot \frac{4}{5} (3 x^{2} + 4)^{\frac{5}{4}} : \frac{1}{2} (3 x^{2} + 4)^{\frac{5}{4}}

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

解答补常数

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

作图

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