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受欢迎的微积分 >

integral of (cos(x))/(sin(x)-cos(x))

解答

\int \frac{cos ( x )}{sin ( x ) - cos ( x )} d x

解答

\frac{1}{2} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 - \frac{1}{2} ln (sec^{2} (\frac{x}{2})) - arctan (tan (\frac{x}{2})) + C

求解步骤

\int \frac{cos ( x )}{sin ( x ) - cos ( x )} d x

使用换元积分法

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

将

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

用部份分式展开:

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

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

\int \frac{u + 1}{2 ( u ^{2} + 2 u - 1 )} d u = \frac{1}{4} ln u^{2} + 2 u - 1

\int \frac{- u - 1}{2 ( u ^{2} + 1 )} d u = \frac{1}{2} (- \frac{1}{2} ln u^{2} + 1 - arctan (u))

= 2 (\frac{1}{4} ln u^{2} + 2 u - 1 + \frac{1}{2} (- \frac{1}{2} ln u^{2} + 1 - arctan (u)))

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

代回

= 2 (\frac{1}{4} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 + \frac{1}{2} (- \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 1 - arctan (tan (\frac{x}{2}))))

化简

2 (\frac{1}{4} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 + \frac{1}{2} (- \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 1 - arctan (tan (\frac{x}{2})))) : \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 - \frac{1}{2} ln (sec^{2} (\frac{x}{2})) - arctan (tan (\frac{x}{2}))

= \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 - \frac{1}{2} ln (sec^{2} (\frac{x}{2})) - arctan (tan (\frac{x}{2}))

解答补常数

= \frac{1}{2} ln tan^{2} (\frac{x}{2}) + 2 tan (\frac{x}{2}) - 1 - \frac{1}{2} ln (sec^{2} (\frac{x}{2})) - arctan (tan (\frac{x}{2})) + C

作图

流行的例子

integral of 1/(2x)

\int \frac{1}{2 x} d x

integral from 1 to 4 of 1/((x-2)^{1/3)}

\int_{1}^{4} \frac{1}{( x - 2 ) ^{\frac{1}{3}}} d x

derivative of 10^{-x}

\frac{d}{d x} (1 0^{- x})

integral of x(8-x)^2

\int x (8 - x)^{2} d x

y^'+sin(x)y=sin(x)

y^{^{'}} + sin (x) y = sin (x)