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受欢迎的微积分 >

integral from 0 to 1.5 of 1/(1+sin(x))

解答

\int_{0}^{1.5} \frac{1}{1 + sin ( x )} d x

解答

0.96458 \dots

求解步骤

\int_{0}^{1.5} \frac{1}{1 + sin ( x )} d x

使用换元积分法

= \int_{0}^{0.93159 \dots} \frac{2}{1 + u ^{2} + 2 u} d u

提出常数:

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

= 2 \cdot \int_{0}^{0.93159 \dots} \frac{1}{1 + u ^{2} + 2 u} d u

对

1 + u^{2} + 2 u

配方:

(u + 1)^{2}

= 2 \cdot \int_{0}^{0.93159 \dots} \frac{1}{( u + 1 ) ^{2}} d u

使用换元积分法

= 2 \cdot \int_{1}^{1.93159 \dots} \frac{1}{v ^{2}} d v

使用幂法则

= 2 [- \frac{1}{v}]_{1}^{1.93159 \dots}

带入上下限计算后得:

0.48229 \dots

= 2 \cdot 0.48229 \dots

化简

= 0.96458 \dots

作图

流行的例子

integral from 0 to 1 of x^3sqrt(7+5x^4)

\int_{0}^{1} x^{3} 7 + 5 x^{4} d x

integral from 0 to 1 of (x-6)e^x

\int_{0}^{1} (x - 6) e^{x} d x

integral from-1 to 2 of (x+4)-(x^2+2)

\int_{- 1}^{2} (x + 4) - (x^{2} + 2) d x

integral from-8 to 5 of (1-x)

\int_{- 8}^{5} (1 - x) d x

integral from 2 to 10 of 1

\int_{2}^{10} 1