Question

$\\ \dagger\ \; \bf Integral\ challenge\ :$ $\\ \displaystyle \bullet\ \; \red{\rm \int \dfrac{1}{x^2(x^4+1)^{\frac{3}{4}}}\ dx}$ ​

Answer 1

Answer:

$\displaystyle \bullet\ \; \purple{\rm \int \dfrac{1}{x^2(x^4+1)^{\frac{3}{4}}}\ dx}$

$\displaystyle \implies\; {\rm \int \dfrac{1}{x^2(x^4+1)^{\frac{3}{4}}}\ dx}$

$\displaystyle \implies\; {\rm \int \dfrac{1}{x^5\:\bigg(1+\dfrac{1}{x^4}\bigg)^{\frac{3}{4}}}\ dx}$

$\displaystyle \implies\; {\rm \int \dfrac{dx}{x^5}\:\dfrac{1}{\bigg(1+\dfrac{1}{x^4}\bigg)^{\frac{3}{4}}}}$

Let,

$\sf{1\:+\:\dfrac{1}{x^4}\:=\:u}$

$\longrightarrow\:\sf{du\:=\:(-4x^{-5})\:.\:dx}$

$\longrightarrow\:\sf{du\:=\:\dfrac{-4}{x^{5}}\:.\:dx}$

$\longrightarrow\:\sf{\dfrac{du}{-4}\:=\:\dfrac{dx}{x^{5}}\:}$

☛ Now putting these value in the above equation given as,

$\displaystyle \implies\; {\rm \int \dfrac{du}{-4}\:\dfrac{1}{u^{\frac{3}{4}}}}$

$\displaystyle \implies\; {\rm \dfrac{1}{-4}\int \dfrac{du}{u^{\frac{3}{4}}}}$

$\displaystyle \implies\; {\rm \dfrac{1}{-4}\int u^{-\frac{3}{4}}\:du}$

$\displaystyle \implies\; {\rm \dfrac{1}{-4} \bigg[\dfrac{u^{-\frac{3}{4}\:+\:1}}{-\frac{3}{4}\:+\:1} \:+\:C\bigg]}$

$\displaystyle \implies\; {\rm \dfrac{1}{-4} \bigg[\dfrac{u^{\frac{1}{4}}}{\frac{1}{4}} \bigg]\:+\:C}$

$\displaystyle \implies\; {\rm \dfrac{1}{-4} \times{4} \: \bigg[u^{\frac{1}{4}}\bigg]\:+\:C}$

$\displaystyle \implies\; {\rm -1 \times{u^{\frac{1}{4}}}\:+\:C}$

$\displaystyle \implies\; {\rm -u^{\frac{1}{4}}\:+\:C}$

$\displaystyle \implies\; \green {\rm -\bigg(1\:+\:\dfrac{1}{x^4}\bigg)^{\frac{1}{4}}\:+\:C}$

Answer 2

${\large{\underbrace{\mathbb{\pink{ ANSWER\: \:-: }}}}}$

$\displaystyle \bullet\ \; \purple{\rm \int \dfrac{1}{x^2(x^4+1)^{\frac{3}{4}}}\ dx}$

$\displaystyle \implies\; {\rm \int \dfrac{1}{x^2(x^4+1)^{\frac{3}{4}}}\ dx}$

$\displaystyle \implies\; {\rm \int \dfrac{1}{x^5\:\bigg(1+\dfrac{1}{x^4}\bigg)^{\frac{3}{4}}}\ dx}$

$\displaystyle \implies\; {\rm \int \dfrac{dx}{x^5}\:\dfrac{1}{\bigg(1+\dfrac{1}{x^4}\bigg)^{\frac{3}{4}}}}$

Let,

$\sf{1\:+\:\dfrac{1}{x^4}\:=\:u}$

$\longrightarrow\:\sf{du\:=\:(-4x^{-5})\:.\:dx}$

$\longrightarrow\:\sf{du\:=\:\dfrac{-4}{x^{5}}\:.\:dx}$

$\longrightarrow\:\sf{\dfrac{du}{-4}\:=\:\dfrac{dx}{x^{5}}\:}$

Now putting these value in the above equation given as,

$\displaystyle \implies\; {\rm \int \dfrac{du}{-4}\:\dfrac{1}{u^{\frac{3}{4}}}}$

$\displaystyle \implies\; {\rm \dfrac{1}{-4}\int \dfrac{du}{u^{\frac{3}{4}}}}$

$\displaystyle \implies\; {\rm \dfrac{1}{-4}\int u^{-\frac{3}{4}}\:du}$

$\displaystyle \implies\; {\rm \dfrac{1}{-4} \bigg[\dfrac{u^{-\frac{3}{4}\:+\:1}}{-\frac{3}{4}\:+\:1} \:+\:C\bigg]}$

$\displaystyle \implies\; {\rm \dfrac{1}{-4} \bigg[\dfrac{u^{\frac{1}{4}}}{\frac{1}{4}} \bigg]\:+\:C}$

$\displaystyle \implies\; {\rm \dfrac{1}{-4} \times{4} \: \bigg[u^{\frac{1}{4}}\bigg]\:+\:C}$

$\displaystyle \implies\; {\rm -1 \times{u^{\frac{1}{4}}}\:+\:C}$

$\displaystyle \implies\; {\rm -u^{\frac{1}{4}}\:+\:C}$

$\displaystyle \implies\; \blue {\rm -\bigg(1\:+\:\dfrac{1}{x^4}\bigg)^{\frac{1}{4}}\:+\:C}$