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Question

If f(y)=ey,g(y)=y,y>0 and F(t)=t0f(ty)g(y)dt, then

A
F(t)=et(1+t)
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B
F(t)=tet
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C
F(t)=tet
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D
F(t)=1et(1+t)
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Solution

The correct option is A F(t)=et(1+t)
We have f(y)=ey,g(y)=y:y>0
F(t)=t0f(ty)g(y)dy
=t0etyydy
=ett0eyydy

using by-parts theorem

=et([yey]t0+t0eydy)=et(tet[ey]t0)
=et(tetet+1)
=et(1+t)

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