JEE Main2003MathematicsDefinite IntegrationActual
If f ( y )= e ^ y , g ( y )= y ; y >0 and F(t)= ₀^ t f ( t - y ) g ( y ) , then
Options
- AF ( t )= te ^ -t
- BF ( t )=1- te ^ -t (1+ t )
- CF ( t )= e ^ t -(1+ t )
- DF ( t )= te ^ t
Correct answer
C. F ( t )= e ^ t -(1+ t )
Step-by-step solution
F(t)= ₀^t f(t-y) g(y) d y= ₀^t e^ t-y y d y=e^t ₀^t e^ -y y d y=e^t [-y e^ -y -e^ -y ]₀^t=-e^t [y e^ -y +e^ -y ]₀^t=-e^t [t e^ -t +e^ -t -0-1 ]=e^t [ t+1-e^t e^t ]=e^t-(1+t)