# Thread: Please check this integral for me

1. ## Please check this integral for me

Hi can sum1 check this trig integral for me please, thanks.

evaluate the integral of

( cot(x) + csc(x) ) / sin(x) dx

I got:

csc(x) (cot(x) + csc(x)) dx

expanding

(csc(x)cot(x)) +csc^2(x) dx

int csc^2(x) dx + int csc(x)cot(x) dx

int csc^2(x) dx + -csc(x)

-cot(x) - csc(x) + C

2. Originally Posted by shadow85
Hi can sum1 check this trig integral for me please, thanks.

evaluate the integral of

( cot(x) + csc(x) ) / sin(x) dx

I got:

csc(x) (cot(x) + csc(x)) dx

expanding

(csc(x)cot(x)) +csc^2(x) dx

int csc^2(x) dx + int csc(x)cot(x) dx

int csc^2(x) dx + -csc(x)

-cot(x) - csc(x) + C
Try taking the derivative of your answer.

3. $\displaystyle \displaystyle \frac{\cot{x} + \csc{x}}{\sin{x}} = \frac{\frac{\cos{x}}{\sin{x}} + \frac{1}{\sin{x}}}{\sin{x}}$

$\displaystyle \displaystyle = \frac{\frac{\cos{x} + 1}{\sin{x}}}{\sin{x}}$

$\displaystyle \displaystyle = \frac{\cos{x} + 1}{\sin^2{x}}$

$\displaystyle \displaystyle = \frac{\cos{x}}{\sin^2{x}} + \frac{1}{\sin^2{x}}$

$\displaystyle \displaystyle = \frac{\cos{x}}{\sin^2{x}} + \frac{\cos^2{x}}{\cos^2{x}\sin^2{x}}$

$\displaystyle \displaystyle = \frac{\cos{x}}{\sin^2{x}} + \frac{\sec^2{x}}{\tan^2{x}}$.

So $\displaystyle \displaystyle \int{\frac{\cot{x} + \csc{x}}{\sin{x}}\,dx} = \int{\frac{\cos{x}}{\sin^2{x}}\,dx} + \int{\frac{\sec^2{x}}{\tan^2{x}}\,dx}$.

Make the substitution $\displaystyle \displaystyle u = \sin{x}$ so that $\displaystyle \displaystyle du = \cos{x}\,dx$ and make the substitution $\displaystyle \displaystyle v = \tan{x}$ so that $\displaystyle \displaystyle dv = \sec^2{x}\,dx$ and the integrals become

$\displaystyle \displaystyle \int{u^{-2}\,du} + \int{v^{-2}\,dv}$

$\displaystyle \displaystyle = \frac{u^{-1}}{-1} + \frac{v^{-1}}{-1} + C$

$\displaystyle \displaystyle = -\frac{1}{\sin{x}} - \frac{1}{\tan{x}} + C$

$\displaystyle \displaystyle = -\csc{x} - \cot{x} + C$.

4. thank you.