Evaluate the following integrals :\(\int\frac{e^{x-1}+x^{e-1}}{e^x+x^e

1.	Evaluate the following integrals :\(\int\frac{e^{x-1}+x^{e-1}}{e^x+x^e}\)dx
Answer» 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.

Evaluate the following integrals :\(\int\frac{e^{x-1}+x^{e-1}}{e^x+x^e}\)dx

Answer»

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.