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Question
Integrate the following functions w.r.t. x : cosec (log x)[1 – cot (log x)]
Solution
Let I = `int "cosec" (log x)[1 - cot (log x)].dx`
Put log x = t
∴ et
∴ dx = et .dt
∴ I = `int "cosec" t (1 - cot t).e^t dt`
= `int e^t ["cosec" t - "cosec" t cot t].dt`
= `int e^t ["cosec" t + d/dt ("cosec" t)].dt`
= `e^t "cosec" t + c ...[∵ int e^t [f(t) + f'(t)].dt = e^t f(t) + c]`
= x . cosec (log x) + c.
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