Question

$\int x^{x^2+1}(2\ln x+1)dx$

Answer 1

Given : $\int x^{x^2+1}(2\ln x+1)dx$

Let :

$t=x^{x^2}{\log }_{e}t=x^2\log x$

Differentiating with respect to $x$

$\frac{1}{t}dt=2\times \ln x+x^2\times \frac{1}{x}dt=x^{x^2+1}(2\ln x+1)dx\int x^{x^2+1}(2\ln x+1)dx=\int dt=t=x^{x^2}+C$

Hence the correct answer is $x^{x^2}+C$