AP EAMCET201823 Apr 2018Morning ShiftMathematicsIndefinite IntegrationActual
^3 x+ ^5 x ^2 x+ ^4 x d x=
Options
- Ax-6 ⁻¹( x)+c
- Bx-2( x)⁻¹+c
- Cx-2( x)⁻¹-6 ⁻¹( x)+c
- Dx-2( x)⁻¹+5 ⁻¹( x)+c
Correct answer
C. x-2( x)⁻¹-6 ⁻¹( x)+c
Step-by-step solution
We have, aligned & = ^3 x+ ^5 x ^2 x+ ^4 x d x & = ^2 x x (1+ ^2 x ) ^2 x (1+ ^2 x ) d x & = (1- ^2 x ) (2- ^2 x ) x ^2 x (1+ ^2 x ) d x aligned Substitute x=t aligned & x d x=d t & = (1-t^2 ) (2-t^2 ) t^2 (1+t^2 ) d t= 2-3 t^2+t^4 t^2 (1+t^2 ) d t & = t^2 (1+t^2 )-4 (t^2+1 )+6 t^2 (1+t^2 ) d t & = (1- 4 t^2 + 6 t^2 (t+t^2 ) ) d t & = 1- 4 t^2 +6 ( 1 t^2 - 1 1+t^2 ) ] d x aligned aligned & = (1+ 2 t^2 - 6 1+t^2 ) d t & =t-2 t⁻¹-6 ⁻¹ t+c & = x-2( x)⁻¹-6 ⁻¹( x)+c aligned