Evaluate the following integrals : ∫(e^(x-1)+x^(e-1))/(e^x+x^e) dx ​

Question

Evaluate the following integrals :

\(\int\frac{e^{x-1}+x^{e-1}}{e^x+x^e}\)dx

Answer 1

Multiplying and dividing the numerator by e we get the given as

⇒ \(\frac{1}{e}\)\(\int\frac{e^{x-1}+x^{e-1}}{e^x+x^6}\)dx  …(1)

Assume,

e^x + x^e = t 

⇒ d(e^x + x^e )= dt 

⇒ e^x + ex^e-1 = dt 

Substituting t and dt in equation 1 we get 

⇒ \(\frac{1}{e}\int\frac{dt}{t}\)

= ln|t| + c 

But,

t = e^x + x^e 

∴ ln| e^x + x^e | + c.