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受欢迎的微积分 >

integral of 1/(3-5sin(x))

解答

\int \frac{1}{3 - 5 sin ( x )} d x

解答

2 (- \frac{1}{8} ln 3 tan (\frac{x}{2}) - 1 + \frac{1}{8} ln tan (\frac{x}{2}) - 3) + C

求解步骤

\int \frac{1}{3 - 5 sin ( x )} d x

使用换元积分法

= \int \frac{2}{3 ( 1 + u ^{2} ) - 10 u} d u

提出常数:

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

= 2 \cdot \int \frac{1}{3 ( 1 + u ^{2} ) - 10 u} d u

将

\frac{1}{3 ( 1 + u ^{2} ) - 10 u}

用部份分式展开:

- \frac{3}{8 ( 3 u - 1 )} + \frac{1}{8 ( u - 3 )}

= 2 \cdot \int - \frac{3}{8 ( 3 u - 1 )} + \frac{1}{8 ( u - 3 )} d u

使用积分加法定则:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= 2 (- \int \frac{3}{8 ( 3 u - 1 )} d u + \int \frac{1}{8 ( u - 3 )} d u)

\int \frac{3}{8 ( 3 u - 1 )} d u = \frac{1}{8} ln ∣ 3 u - 1 ∣

\int \frac{1}{8 ( u - 3 )} d u = \frac{1}{8} ln ∣ u - 3 ∣

= 2 (- \frac{1}{8} ln ∣ 3 u - 1 ∣ + \frac{1}{8} ln ∣ u - 3 ∣)

u = tan (\frac{x}{2})

代回

= 2 (- \frac{1}{8} ln 3 tan (\frac{x}{2}) - 1 + \frac{1}{8} ln tan (\frac{x}{2}) - 3)

解答补常数

= 2 (- \frac{1}{8} ln 3 tan (\frac{x}{2}) - 1 + \frac{1}{8} ln tan (\frac{x}{2}) - 3) + C

作图

流行的例子

麦克劳林 (1+x)^{-1}

ma c l a u r in (1 + x)^{- 1}

integral of (2x)/(x^2-16)+1/x+2

\int \frac{2 x}{x ^{2} - 16} + \frac{1}{x} + 2 d x

y^{^{'}} = x

integral from 0 to 1 of 8e^{-4x}

\int_{0}^{1} 8 e^{- 4 x} d x

integral of-y/x

\int - \frac{y}{x} d x