Discussion Forum



1)

The integral  $\int_{}^{} \frac{dx}{x^{2}(x^{4}+1)^{\frac{3}{4}}}$  equals


A) $\left[ \frac{x^{4}+1}{x^{4}}\right]^{\frac{1}{4}}+C$

B) $(x^{4}+1)^{\frac{1}{4}}+C$

C) $-(x^{4}+1)^{\frac{1}{4}}+C$

D) $-\left[ \frac{x^{4}+1}{x^{4}}\right]^{\frac{1}{4}}+C$

Answer:

Option D

Explanation:

$\int_{}^{} \frac{dx}{x^{2}(x^{4}+1)^{\frac{3}{4}}}$   = $\int_{}^{} \frac{dx}{x^{5}(1+\frac{1}{x^{4}})^{\frac{3}{4}}}$

 Put            $1+\frac{1}{x^{4}}=t^{4}$

    $\Rightarrow$        $-\frac{4}{x^{5}}dx=4t^{3}dt$

    $\Rightarrow$     $\frac{dx}{x^{5}}=-t^{3}dt=\int_{}^{} \frac{-t^{3}dt}{t^{3}}$

  =   $-\int_{}^{} dt=-t+C=-\left(1+\frac{1}{x^{4}}\right)^{\frac{1}{4}}+c$