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Question
Integrate the following functions w.r.t.x:
`e^(5x).[(5x.logx + 1)/x]`
Solution
Let I = `int e^(5x) [(5x.log x + 1)/x].dx`
= `int e^(5x)[5log x + 1/x].dx`
Put 5x = t
∴ 5.dx = dt
∴ dx = `(1)/(5).dt`
Also, x = `t/(5)`
∴ I = `(1)/(5) int e^t [5 log (t/5) + 5/t].dt`
Let f(t) = `5log (t/5)`
= 5 log t – 5 log 5
∴ f'(t) = `d/dt [5log t - 5 log 5]`
= `(5)/t - 0`
= `(5)/t`
∴ I = `(1)/(5) int e^t [f(t) + f^'(t)].dt`
= `(1)/(5) e^t f(t) + c`
= `(1)/(5) e^t . 5log (t/5) + c`
= e5x log x + c.
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