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受欢迎的微积分 >

integral of 36t(9t^2-7)^3

解答

\int 36 t (9 t^{2} - 7)^{3} d t

解答

\frac{1}{2} (9 t^{2} - 7)^{4} + C

求解步骤

\int 36 t (9 t^{2} - 7)^{3} d t

提出常数:

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

= 36 \cdot \int t (9 t^{2} - 7)^{3} d t

使用换元积分法

= 36 \cdot \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

= 36 \cdot \frac{1}{18} \cdot \int u^{3} d u

使用幂法则

= 36 \cdot \frac{1}{18} \cdot \frac{u ^{4}}{4}

u = 9 t^{2} - 7

代回

= 36 \cdot \frac{1}{18} \cdot \frac{( 9 t ^{2} - 7 ) ^{4}}{4}

化简

36 \cdot \frac{1}{18} \cdot \frac{( 9 t ^{2} - 7 ) ^{4}}{4} : \frac{1}{2} (9 t^{2} - 7)^{4}

= \frac{1}{2} (9 t^{2} - 7)^{4}

解答补常数

= \frac{1}{2} (9 t^{2} - 7)^{4} + C

作图

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