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Evaluate the integral of : cosecx*cotX all devide by 2-cosecX dx?

1 Answer

Apr 28, 2018

$I = ln (| 2 - csc (x) |) + C$ $I=ln(abs(2-csc(x)))+C$

Explanation:

We want to integrate

$I = \int \frac{csc (x) cot (x)}{2 - csc (x)} d x$ $I=int(csc(x)cot(x))/(2-csc(x))dx$

Make a substitution $u = 2 - csc (x) \Rightarrow d u = cot (x) csc (x) d x$ $color(red)(u=2-csc(x)=>du=cot(x)csc(x)dx$

$I = \int \frac{1}{u} d u$ $I=int1/udu$

$I = ln (| u |) + C$ $color(white)(I)=ln(abs(u))+C$

Substitute back $u = 2 - csc (x)$ $color(red)(u=2-csc(x)$

$I = ln (| 2 - csc (x) |) + C$ $I=ln(abs(2-csc(x)))+C$

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