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受欢迎的微积分 >

integral of (sin(x))/(cos^2(x)+cos(x)-6)

解答

\int \frac{sin ( x )}{cos ^{2} ( x ) + cos ( x ) - 6} d x

解答

- 2 (\frac{1}{10} ln 3 tan^{2} (\frac{x}{2}) + 1 - \frac{1}{10} ln tan^{2} (\frac{x}{2}) + 2) + C

求解步骤

\int \frac{sin ( x )}{cos ^{2} ( x ) + cos ( x ) - 6} d x

使用三角恒等式改写

= \int \frac{2 sin ( x )}{cos ( 2 x ) + 2 cos ( x ) - 11} d x

提出常数:

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

= 2 \cdot \int \frac{sin ( x )}{cos ( 2 x ) + 2 cos ( x ) - 11} d x

使用三角恒等式改写

= 2 \cdot \int \frac{sin ( x )}{2 cos ^{2} ( x ) + 2 cos ( x ) - 12} d x

使用换元积分法

= 2 \cdot \int - \frac{u}{3 u ^{4} + 7 u ^{2} + 2} d u

提出常数:

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

= 2 (- \int \frac{u}{3 u ^{4} + 7 u ^{2} + 2} d u)

将

\frac{u}{3 u ^{4} + 7 u ^{2} + 2}

用部份分式展开:

\frac{3 u}{5 ( 3 u ^{2} + 1 )} - \frac{u}{5 ( u ^{2} + 2 )}

= 2 (- \int \frac{3 u}{5 ( 3 u ^{2} + 1 )} - \frac{u}{5 ( u ^{2} + 2 )} 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 u}{5 ( 3 u ^{2} + 1 )} d u - \int \frac{u}{5 ( u ^{2} + 2 )} d u))

\int \frac{3 u}{5 ( 3 u ^{2} + 1 )} d u = \frac{1}{10} ln 3 u^{2} + 1

\int \frac{u}{5 ( u ^{2} + 2 )} d u = \frac{1}{10} ln u^{2} + 2

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

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

代回

= 2 (- (\frac{1}{10} ln 3 tan^{2} (\frac{x}{2}) + 1 - \frac{1}{10} ln tan^{2} (\frac{x}{2}) + 2))

化简

= - 2 (\frac{1}{10} ln 3 tan^{2} (\frac{x}{2}) + 1 - \frac{1}{10} ln tan^{2} (\frac{x}{2}) + 2)

解答补常数

= - 2 (\frac{1}{10} ln 3 tan^{2} (\frac{x}{2}) + 1 - \frac{1}{10} ln tan^{2} (\frac{x}{2}) + 2) + C

作图

流行的例子

tangent f(x)=x^2-3x+5,\at x=1

t an g e n t f (x) = x^{2} - 3 x + 5, at x = 1

sum from n=0 to infinity of n^3

n = 0 \sum \infty n^{3}

tangent f(x)=-7x^3-8x^2-5x,\at x=-3

t an g e n t f (x) = - 7 x^{3} - 8 x^{2} - 5 x, at x = - 3

integral of (sin(2x))/(1+cos(2x))

\int \frac{sin ( 2 x )}{1 + cos ( 2 x )} d x

integral from 1 to 2 of e^{1/x}

\int_{1}^{2} e^{\frac{1}{x}} d x