Maharashtra BoardEnglish MediumSTD 12 Commerce / ArtsMaths (commerce)Integration (p-1)3 Marks
Question
Evalute : $\int \frac{1}{x\left(x^6+1\right)} d x$
✓
Answer
Let $I=\int \frac{1}{x\left(x^6+1\right)} d x$ $ =\int \frac{x^5}{x^6\left(x^6+1\right)} d x $ Put $x^6=t \quad \therefore 6 x^5 d x=d t$ $ \therefore x^5 d x=\frac{1}{6} d t $ $ \begin{aligned} \therefore I & =\int \frac{1}{t(t+1)} \cdot \frac{d t}{6} \\ & =\frac{1}{6} \int \frac{(t+1)-t}{t(t+1)} d t=\frac{1}{6} \int\left(\frac{1}{t}-\frac{1}{t+1}\right) d t \\ & =\frac{1}{6}\left[\int \frac{1}{t} d t-\int \frac{1}{t+1} d t\right] \\ & =\frac{1}{6}[\log (t)-\log |t+1|]+c \\ & =\frac{1}{6} \log \left|\frac{t}{t+1}\right|+c=\frac{1}{6} \log \left|\frac{x^6}{x^6+1}\right|+c . \end{aligned} $
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