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受欢迎的微积分 >

integral from 0 to pi/4 of 1/(1-sin(x))

解答

\int_{0}^{\frac{π}{4}} \frac{1}{1 - sin ( x )} d x

解答

2

+1

十进制

1.41421 \dots

求解步骤

\int_{0}^{\frac{π}{4}} \frac{1}{1 - sin ( x )} d x

使用换元积分法

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

对

1 + u^{2} - 2 u

配方:

(u - 1)^{2}

= 2 \cdot \int_{0}^{2 - 1} \frac{1}{( u - 1 ) ^{2}} d u

使用换元积分法

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

使用幂法则

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

带入上下限计算后得:

\frac{1 + 2}{2} - 1

= 2 (\frac{1 + 2}{2} - 1)

化简

= 2

作图

流行的例子

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

integral from-pi/2 to pi/2 of cos(x)-2

\int_{- \frac{π}{2}}^{\frac{π}{2}} cos (x) - 2 d x

integral from 0 to 6 of e^{-1/3 t}

\int_{0}^{6} e^{- \frac{1}{3} t} d t

integral from 0 to x of 2(1-x)

\int_{0}^{x} 2 (1 - x) d x

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\int_{a}^{b} x (\frac{1}{b - a}) d x