MHT CET202619 April 2026Morning ShiftMathematicsIndefinite IntegrationActual
The value of integral x^2 + 1 ,[ (x^2 + 1) - 2 x] x^4 ,dx is equal to...
Options
- A(1 + 1 x^2 )^ 3/2 [ -1 3 (1 + 1 x^2 ) + 2 9 ] + c
- B(1 + 1 x^2 )^ 3/2 [ -1 3 (1 + 1 x^2 ) - 2 9 ] + c
- C(1 + 1 x^2 )^ 3/2 [ -1 3 (1 + 1 x^2 ) + 2 3 ] + c
- D(1 + 1 x^2 )^ 3/2 [ -1 3 (1 + 1 x^2 ) - 2 3 ] + c
Correct answer
A. (1 + 1 x^2 )^ 3/2 [ -1 3 (1 + 1 x^2 ) + 2 9 ] + c
Step-by-step solution
Let the given integral be I = x^2+1 [ (x^2+1) - 2 x] x^4 dx I = x^2(1+ 1 x^2 ) x^4 [ (x^2+1) - x^2] dx I = x 1+ 1 x^2 x^4 ( x^2+1 x^2 ) dx I = 1 x^3 1+ 1 x^2 (1+ 1 x^2 ) dx Substituting 1+ 1 x^2 = t Differentiating with respect to x , we get - 2 x^3 dx = dt 1 x^3 dx = - 1 2 dt I = t t (- 1 2 ) dt I = - 1 2 t^ 1/2 t dt Using integration by parts, taking t as the first function and t^ 1/2 as the second function: I = - 1 2 [ t t^ 1/2 dt - ( d dt ( t) t^ 1/2 dt ) dt ] I = - 1 2 [ t ( t^ 3/2 3/2 ) - 1 t ( t^ 3/2 3/2 ) d