Evaluate the following functions : 1 3 x+7 x^ -n d x - Vidyadip

Question

Evaluate the following functions : $\int \frac{1}{3 x+7 x^{-n}} \cdot d x$

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Answer

$
\begin{aligned}
& \text { : Consider } \int \frac{1}{3 x+7 x^{-n}} \cdot d x \\
& \quad=\int \frac{1}{3 x+\frac{7}{x^n}} \cdot d x=\int \frac{1}{\frac{3 x^{n+1}+7}{x^n}} \cdot d x \\
& \quad=\int \frac{x^n}{3 x^{n+1}+7} \cdot d x \\
& \quad \text { put } 3 x^{n+1}+7=t \\
& \quad \text { Differentiate } w \cdot r \cdot t \cdot x \\
& \quad 3(n+1) x^n \cdot d x=d t \\
& \therefore \quad x^n \cdot d x=\frac{1}{3(n+1)} d t \\
& =\int \frac{1}{3(n+1)} \cdot d t \\
& =\frac{1}{3(n+1)} \cdot \log (t)+c \\
& =\frac{1}{3(n+1)} \cdot \log \left(3 x^{n+1}+7\right)+c
\end{aligned}
$

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