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受欢迎的微积分 >

integral from-1 to 0 of (2t)/((2+t^2)^3)

解答

\int_{- 1}^{0} \frac{2 t}{( 2 + t ^{2} ) ^{3}} d t

解答

- \frac{5}{72}

+1

十进制

- 0.06944 \dots

求解步骤

\int_{- 1}^{0} \frac{2 t}{( 2 + t ^{2} ) ^{3}} d t

提出常数:

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

= 2 \cdot \int_{- 1}^{0} \frac{t}{( 2 + t ^{2} ) ^{3}} d t

使用换元积分法

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

\int_{a}^{b} f (x) d x = - \int_{b}^{a} f (x) d x, a < b

= 2 (- \int_{2}^{3} \frac{1}{2 u ^{3}} d u)

提出常数:

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

= 2 (- \frac{1}{2} \cdot \int_{2}^{3} \frac{1}{u ^{3}} d u)

使用幂法则

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

化简

2 (- \frac{1}{2} [- \frac{1}{2 u ^{2}}]_{2}^{3}) : - [- \frac{1}{2 u ^{2}}]_{2}^{3}

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

带入上下限计算后得:

\frac{5}{72}

= - \frac{5}{72}

作图

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