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受欢迎的微积分 >

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

解答

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

解答

\frac{5}{12} (2 x^{2} + 3)^{3} + C

求解步骤

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

提出常数:

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

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

使用换元积分法

= 5 \cdot \int \frac{u ^{2}}{4} d u

提出常数:

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

= 5 \cdot \frac{1}{4} \cdot \int u^{2} d u

使用幂法则

= 5 \cdot \frac{1}{4} \cdot \frac{u ^{3}}{3}

u = 2 x^{2} + 3

代回

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

化简

5 \cdot \frac{1}{4} \cdot \frac{( 2 x ^{2} + 3 ) ^{3}}{3} : \frac{5}{12} (2 x^{2} + 3)^{3}

= \frac{5}{12} (2 x^{2} + 3)^{3}

解答补常数

= \frac{5}{12} (2 x^{2} + 3)^{3} + C

作图

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