\(\rm\displaystyle\int\frac{d x}{x\left(x^5+3\right)}\) is equal to

Question

Question

Download Solution PDF

\(\rm\displaystyle\int\frac{d x}{x\left(x^5+3\right)}\) is equal to

This question was previously asked in

CUET Mathematics 30th Aug 2022 Official Paper

Attempt Online

View all CUET Papers >

\(\rm\frac{1}{3} \log \left| \frac{x^5}{x^5+3}\right|+C\)
\(\rm\frac{1}{15} \log \left|\frac{x^5}{x^5+3}\right|+ C\)
\(\rm\frac{1}{5} \log \left|\frac{x^5}{x^5+3}\right|+C\)
\(\rm\frac{1}{25}\log\left|\frac{x^5}{x^5+3}\right|+C\)

Answer 1

Calculations:

∫1/x((x⁵)+3)dx

To solve this integral, we can use the substitution method. Let:

u = x⁵+3

du = 5x⁴dx

∫1/x((x5)+3)dx = ∫1/u((5x⁴))du = 1/5 [∫1/u((u-3))]du

By partial Fraction

1/5 ∫(A(u-3) + Bu)/u(u-3) du =1/5 ( -1/3 ∫1/u du + 1/3 ∫1/(u - 3))du

= 1/15 (Log u-3)/log u = \(\frac{1}{5}\) \(\rm\frac{1}{3} \log \left| \frac{x^5}{x^5+3}\right|+C\) = \(\rm\frac{1}{15} \log \left|\frac{x^5}{x^5+3}\right|+ C\)

Download Solution PDF

Question

\(\rm\displaystyle\int\frac{d x}{x\left(x^5+3\right)}\) is equal to

Answer (Detailed Solution Below)

Detailed Solution

More Mathematics Questions