MHT CET20244 May 2024Evening ShiftMathematicsDefinite IntegrationActual
The value of _ 6 ^ 4 1 2 x ( ^5 x+ ^5 x ) dx is
Options
- A1 5 ( 4 - ⁻¹ ( 1 3 3 ) )
- B1 2 ( 4 - ⁻¹ ( 1 9 3 ) )
- C1 10 ( 4 - ⁻¹ ( 1 9 3 ) )
- D1 10 ( 4 - ⁻¹ ( 1 3 3 ) )
Correct answer
C. 1 10 ( 4 - ⁻¹ ( 1 9 3 ) )
Step-by-step solution
aligned & I= _ 6 ^ 4 1 2 x ( ^5 x+ ^5 x ) d x & = _ 6 ^ 4 1 2 x 1+ ^2 x ( ^5 x+ 1 ^5 x ) d x & = _ 6 ^ 4 (1+ ^2 x ) 2 x ( ^5 x+ 1 ^5 x ) d x aligned I= _ 6 ^ 4 ^2 x 2 x ( ^5 x+ 1 ^5 x ) d x Let x= t ^2 x ~d x= dt when x= 6 , t = 1 3 when x= 4 , t=1 array ll & I= _ 1 3 ^1 d t 2 t (t^5+ 1 t^5 ) & I= _ 1 3 ^1 t^4 2 (t¹⁰+1 ) d t & Let t^5=u & 5 t^4 d t=d u array When t = 1 3 , u =3^ -5 2 When t =1, u =1 aligned I & = 1 10 _ -5 32 ^1 d u (u^2+1 ) & = 1 10 ( ⁻¹ u )_ -5 2 ^1 & = 1 10 [ ⁻¹ 1- ⁻¹ (3^ -5 2 ) ] & = 1 10 [ 4 -