Evaluate the following integrals :\(\int\)\(5^{x+tan^{-1}x}\)(\(\frac{

1.	Evaluate the following integrals :\(\int\)\(5^{x+tan^{-1}x}\)(\(\frac{x^2+2}{x^2+1}\))dx
Answer» Assume, x + tan^-1x = t d(x + tan^-1x) = dt ⇒ 1 + \(\frac{1}{x^2+1}\) = dt ⇒ \(\frac{2+x^2}{x^2+1}\) = dt Substituting t and dt ⇒ \(\int\)5^tdt ⇒ \(\frac{5^t}{log\,5}\) + c But, t = x + tan^-1x ⇒ \(\frac{5^{x+tan^{-1}x}}{log\,5}\) + c.

Evaluate the following integrals :\(\int\)\(5^{x+tan^{-1}x}\)(\(\frac{x^2+2}{x^2+1}\))dx

Answer»

Assume,

x + tan^-1x = t

d(x + tan^-1x) = dt

⇒ 1 + \(\frac{1}{x^2+1}\) = dt

⇒ \(\frac{2+x^2}{x^2+1}\) = dt

Substituting t and dt

⇒ \(\int\)5^tdt

⇒ \(\frac{5^t}{log\,5}\) + c

But,

t = x + tan^-1x

⇒ \(\frac{5^{x+tan^{-1}x}}{log\,5}\) + c.