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KCET2015MathematicsVector Algebra

( 1 x² (x⁴+1 )^ 3 / 4 dx ) is equal to

Options

  1. A( - (1+x⁴ )^ 1 / 4 x +C )
  2. B( - (1+x⁴ )^ 1 / 4 x² +C )
  3. C( - (1+x⁴ )^ 1 / 4 2 x +C )
  4. D( - (1+x⁴ )^ 3 / 4 x +C )

Correct answer

A. ( - (1+x⁴ )^ 1 / 4 x +C )

Step-by-step solution

Given that ( I= d x x² (x⁴+1 )^ 3 / 4 ) ( = d x x² [x⁴ (1+ 1 x⁴ ) ]^ 3 / 4 ) ( = d x x² x³ (1+x⁻⁴ )^ 3 / 4 = x⁻⁵ (1+x⁻⁴ )^ 3 / 4 d x ) Let ( 1+x⁻⁴=t⁴ ) then ( -4 x⁵ d x=4 t³ d t ) So, ( I=- 1 4 4 t³ t³ d t=- d t=-t ) ( =- (1+ 1 x⁴ )^ 1 / 4 =- ( 1+x⁴ x⁴ )^ 1 / 4 =- (1+x⁴ )^ 1 / 4 x +C )

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