Question

JEE ADVANCE :-

Find the value of

$\int \: \frac{ {x}^{4} + 1 }{ {( {x}^{4} + 2)}^{ \frac{3}{4} } } \: dx$
​

Answer 1

To integrate :-

$</p><p></p><p>\int \: \frac{ {x}^{4} + 1 }{ {( {x}^{4} + 2)}^{ \frac{3}{4} } } \: dx </p><p>.</p><p>$

Concept

This type of integration is frequently asked in JEE Advance. It uses critical analytical skills.

The main concepts are :-

• Multiply numerator by denominator by such a factor that the numerator becomes the differentiation of the numerator.

Solution

$</p><p></p><p>\int \: \frac{ {x}^{4} + 1 }{ {( {x}^{4} + 2)}^{ \frac{3}{4} } } \: dx \\ \\ multiplying \: \: num \: and \: denominator \: by \: {x}^{3} \\ \\ = > \int \: \frac{ {x}^{7} + {x}^{3} }{ {( {x}^{8} + 2 {x}^{4} )}^{ \frac{3}{4} } } dx \\ \\ \\putting \: {x}^{8} + 2 {x}^{4} = t\: \\ \\ we \: get \: (8 {x}^{7} + 8 {x}^{3} ) \: dx=d t \\ \\ ({x}^{7} + {x}^{3} ) \: dx= \frac{dt}{8} \: \\ \\ = > \int \frac{dt}{8 {t}^{ \frac{3}{4} } } \\ \\ \\ = > \frac{4 {t}^{ \frac{1}{4} } }{8} \\ \\ = > \frac{ {t}^{ \frac{1}{4} } }{2} \\ \\ = > \frac{ { ({x}^{8} + 2 {x}^{4} ) }^{ \frac{1}{4} } }{2} </p><p></p><p>$

Hence, the value is

$\boxed{ \: \pink{\frac{ { ({x}^{8} + 2 {x}^{4} ) }^{ \frac{1}{4} } }{2} } \: }</p><p>$