Question

∫[{e^xlog(ex^x)}/x]dx = _______ + c 

(a) (e^x/x)logx^x 

(b) e^xlogx 

(c) e^xxlogx 

(d) log(xe^x)

Answer 1

The correct option  (b) e^xlogx

Explanation:

I = ∫[{e^x ∙ log(ex^x)}/x]dx 

= ∫(e^x/x)[log_ee + xlog_ex]dx 

= ∫(e^x/x)(1 + xloge^x)dx 

I = ∫e^x[(1/x) + loge^x]dx 

I is of the form, ∫e^x[f(x) + f'(x)], 

Whose solution is e^x f(x) + c

∴ by comparing, 

I = e^x ∙ logx + c