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

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Solution

Given : $\int x^{x^{2} + 1} (2 ln x + 1) d x$
Let :
$t = x^{x^{2}} {log}_{e} t = x^{2} log x$
Differentiating with respect to $x$
$\frac{1}{t} d t = 2 \times ln x + x^{2} \times \frac{1}{x} d t = x^{x^{2} + 1} (2 ln x + 1) d x \int x^{x^{2} + 1} (2 ln x + 1) d x = \int d t = t = x^{x^{2}} + C$
Hence the correct answer is $x^{x^{2}} + C$

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