Evaluate: 10 x^9+10^x _e 10 10^x+x^ 10 d x - Vidyadip

Question

Evaluate: $\int \frac{10 x^9+10^x \log _e 10}{10^x+x^{10}} d x$

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Answer

(b) : Let $I=\int \frac{10 x^9+10^x \log _e 10}{10^x+x^{10}} d x$ Put $10^x+x^{10}=t$
$
\begin{aligned}
\Rightarrow \quad & \left(10^x \log _e 10+10 x^9\right) d x=d t \\
\therefore \quad & I=\int \frac{10 x^9+10^x \log _e 10}{10^x+x^{10}} d x=\int \frac{d t}{t} \\
& \quad=\log _e t+C=\log _e\left(10^x+x^{10}\right)+C
\end{aligned}
$

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