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Question
Integrate the following functions w.r.t. x : `(x.sec^2(x^2))/sqrt(tan^3(x^2)`
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Solution
Let I = `int (x.sec^2(x^2))/sqrt(tan^3(x^2)).dx`
Put tan(x2) = t
∴ sec2(x2) x 2x dx = dt
∴ `x.sec^2(x^2)dx = dt/(2)`
∴ I = `int (1)/sqrt(t^3).dt/(2)`
= `(1)/(2) int t^(-3/2)dt`
= `(1)/(2).(t^(-1/2))/(-1/2) + c`
= `(-1)/sqrt(t) + c`
= `(-1)/sqrt(tan(x^2)) + c`.
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