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Integrate (1)/(1-cot x) ​...

Integrate `(1)/(1-cot x) ​

Text Solution

Verified by Experts

The correct Answer is:
`(1)/(2)"log"("sin x" - "cos x")+(x)/(2)+c`
Let `I=int(sin x)/(sin x-cos x)dx`
Again, let sin x = A(cos x + sin x)+B(sin x - cos x), then A+B =1 and A-B = 0
`rArr" "A=(1)/(2), B=(1)/(2)`
`therefore" "I=int((1)/(2)(cos x+sin x)+(1)/(2)(sin x-cos x))/((sin x - cos x))dx`
`=(1)/(2)int(cos x + sin x)/(sin x - cos x)dx + (1)/(2) int 1 dx + c`
`=(1)/(2)"log"(sin x - cos x)+(1)/(2)x + c`
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