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受欢迎的微积分 >

integral of (pix^3+1)^43pix^2

解答

\int (π x^{3} + 1)^{4} 3 π x^{2} d x

解答

\frac{1}{5} (π x^{3} + 1)^{5} + C

求解步骤

\int (π x^{3} + 1)^{4} \cdot 3 π x^{2} d x

提出常数:

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

= 3 π \cdot \int (π x^{3} + 1)^{4} x^{2} d x

使用换元积分法

= 3 π \cdot \int \frac{u ^{4}}{3 π} d u

提出常数:

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

= 3 π \frac{1}{3 π} \cdot \int u^{4} d u

使用幂法则

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

u = π x^{3} + 1

代回

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

化简

3 π \frac{1}{3 π} \cdot \frac{( π x ^{3} + 1 ) ^{5}}{5} : \frac{1}{5} (π x^{3} + 1)^{5}

= \frac{1}{5} (π x^{3} + 1)^{5}

解答补常数

= \frac{1}{5} (π x^{3} + 1)^{5} + C

作图

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