Discussion Forum



1)

$\int\frac{1}{x^{2}(x^{4}+1)^{3/4}}dx$   is equal to


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

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

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

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

Answer:

Option D

Explanation:

$\int  \frac{dx}{x^{2}(x^{4}-1)^{3/4}}=\int \frac{dx}{x^{5}\left(1+\frac{1}{x^{4}}\right)^{3/4}}$

 Put   $1+\frac{1}{x^{4}}=t\Rightarrow-\frac{4}{x^{5}}dx=dt$

 So, integral is 

$I= -\frac{1}{4}\int\frac{dt}{t^{3/4}}=-t^{\frac{1}{4}}+c=-\left(1+\frac{1}{x^{4}}\right)^{1/4}+c$