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Integration (p-1) — Maths (commerce) STD 12 Commerce / Arts — Question

Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Integration (p-1)3 Marks
Question
Evalute : $\int \frac{1}{x\left(x^n+1\right)} d x$
✓

Answer

Let $I=\int \frac{1}{x\left(x^n+1\right)} d x$
$
=\int \frac{x^{n-1}}{x^n\left(x^n+1\right)} d x
$
Put $x^n=t \quad \therefore n x^{n-1} d x=d t$
$
\begin{aligned}
& \therefore x^{n-1} d x=\frac{d t}{n} \\
& \begin{aligned}
\therefore I & =\int \frac{1}{t(t+1)} \cdot \frac{d t}{n} \\
& =\frac{1}{n} \int \frac{(t+1)-t}{t(t+1)} d t \\
& =\frac{1}{n} \int\left(\frac{1}{t}-\frac{1}{t+1}\right) d t \\
& =\frac{1}{n}\left[\int \frac{1}{t} d t-\int \frac{1}{t+1} d t\right] \\
& =\frac{1}{n}[\log |t|-\log |t+1|]+c \\
& =\frac{1}{n} \log \left|\frac{t}{t+1}\right|+c=\frac{1}{n} \log \left|\frac{x^n}{x^n+1}\right|+c .
\end{aligned}
\end{aligned}
$

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