Aspire Faculty ID #17238 · Topic: CUET UG 2022 Applied Mathematics · Just now
CUET UG 2022 Applied Mathematics

$\int \frac{dx}{x(x^5+3)}$ is equal to

Solution

Let

$I=\int \frac{dx}{x(x^5+3)}$

Put $t=x^5$

Then

$dt=5x^4 dx$

Rewrite integral using substitution:

$I=\frac{1}{5}\int \frac{dt}{t(t+3)}$

Now use partial fractions:

$\frac{1}{t(t+3)}=\frac{1}{3}\left(\frac{1}{t}-\frac{1}{t+3}\right)$

So,

$I=\frac{1}{5}\cdot\frac{1}{3}\int\left(\frac{1}{t}-\frac{1}{t+3}\right)dt$

$I=\frac{1}{15}\log\left|\frac{t}{t+3}\right|+C$

Substitute back $t=x^5$

$I=\frac{1}{15}\log\left|\frac{x^5}{x^5+3}\right|+C$
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