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受欢迎的微积分 >

integral from 0 to 1 of 5/((2x+1)^2)

解答

\int_{0}^{1} \frac{5}{( 2 x + 1 ) ^{2}} d x

解答

\frac{5}{3}

+1

十进制

1.66666 \dots

求解步骤

\int_{0}^{1} \frac{5}{( 2 x + 1 ) ^{2}} d x

提出常数:

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

= 5 \cdot \int_{0}^{1} \frac{1}{( 2 x + 1 ) ^{2}} d x

使用换元积分法

= 5 \cdot \int_{1}^{3} \frac{1}{2 u ^{2}} d u

提出常数:

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

= 5 \cdot \frac{1}{2} \cdot \int_{1}^{3} \frac{1}{u ^{2}} d u

使用幂法则

= 5 \cdot \frac{1}{2} [- \frac{1}{u}]_{1}^{3}

化简

5 \cdot \frac{1}{2} [- \frac{1}{u}]_{1}^{3} : \frac{5}{2} [- \frac{1}{u}]_{1}^{3}

= \frac{5}{2} [- \frac{1}{u}]_{1}^{3}

带入上下限计算后得:

\frac{2}{3}

= \frac{5}{2} \cdot \frac{2}{3}

化简

= \frac{5}{3}

作图

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