Question

Evaluate: $\int \frac{dx}{x^4\left[x\right(x^5-1){]}^{\frac{1}{3}}}$

• A. $\frac{3}{2}{\left\{\frac{x^5-1}{x^5}\right\}}^{\frac{2}{3}}+c$
• B. $\frac{3}{10}{\left\{\frac{x^5-1}{x^5}\right\}}^{\frac{2}{3}}+c$
• C. $\frac{3}{4}{\left\{\frac{x^5-1}{x^5}\right\}}^{\frac{2}{3}}+c$
• D. $\frac{3}{5}{\left\{\frac{x^5-1}{x^5}\right\}}^{\frac{2}{3}}+c$

Answer 1

Answer: (B)

$\frac{3}{10}{\left\{\frac{x^5-1}{x^5}\right\}}^{\frac{2}{3}}+c$

$\int \frac{dx}{x^4\left[x\right(x^5-1){]}^{\frac{1}{3}}}$
$I=\int \frac{dx}{x^4{x}^{\frac{1}{3}}(x^5-1{)}^{\frac{1}{3}}}$
$I=\int \frac{dx}{x^4{x}^{\frac{1}{3}}{x}^{\frac{5}{3}}{(1-\frac{1}{x^5})}^{\frac{1}{3}}}$
$I=\int \frac{dx}{x^6{(1-\frac{1}{x^5})}^{\frac{1}{3}}}$
Put $1-\frac{1}{x^5}=t$
$\Rightarrow \frac{5}{x^6}dx=dt$
$I=\frac{1}{5}\int \frac{dt}{t^{\frac{1}{3}}}$
$\Rightarrow I=\frac{3}{10}{t}^{\frac{2}{3}}+C$
$\Rightarrow I=\frac{3}{10}{(1-\frac{1}{x^5})}^{\frac{2}{3}}+C$
$\Rightarrow I=\frac{3}{10}{\left(\frac{x^5-1}{x^5}\right)}^{\frac{2}{3}}+C$