AP EAMCET202317 May 2023Morning ShiftMathematicsIndefinite IntegrationActual
(x^ 3 m +x^ 2 m +x^m ) (2 x^ 2 m +3 x^m+6 )^ 1 m d x=
Options
- A1 6(m+1) (2 x^ 3 m +3 x^ 2 m +6 x^m )^ m+1 m +C
- B1 6(m+1) (2 x^ 3 m +3 x^ 2 m +6 x^m )^ m-1 m +C
- C1 6(m+1) (2 x^ 3 m +3 x^ 2 m +6 )^ m+1 m +C
- D1 6(m-1) (2 x^ 3 m +m x^ 2 m +6 x^m )^ m-1 m +C
Correct answer
A. 1 6(m+1) (2 x^ 3 m +3 x^ 2 m +6 x^m )^ m+1 m +C
Step-by-step solution
aligned & Let I = (x^ 3 x +x^ 2 m +x^m ) (2 x^ 2 m +3 x m+6 )^ 1 m d x & = (x^ 3 m +x^ 2 m +x^m ) ( (2 x^ 2 m +3 x^m+6 ) x^m x^m )^ 1 m d x & = (x^ 3 m-1 +x^ 2 m-1 +x^ m-1 ) (2 x^ 2 m +3 x^ 2 m +6 x^m )^ 1 m d x aligned Put 2 x^ 2 m +3 x^ 2 m +6 x^m=t aligned & (6 m x^ 3 m-1 +6 m x^ 2 m-1 +6 m x^ m-1 ) d x=d t & (x^ 3 m-1 +x^ 2 m-1 +x^ m-1 ) d x= d t 6 m & I = 1 6 m t^ 1 m d t= 1 6 m t^ 1 m +1 ( 1 m +1 ) +c & I = (2 x^ 3 m +3 x^ 2 m +6 x^m ) 6(m+1) +c aligned