Second mean value theorem in Bonnet's form

[Suvadip]

Using second mean value theorem in Bonnet's form show that there exists a
\(\displaystyle p \)in \(\displaystyle [a,b]\) such that
\(\displaystyle \int_a^b e^{-x}cos x dx =sin ~p\)

I know the theorem but how to show this using that theorem .

 

[Jhoneine]

We can solve this by using Mean Value Theorem for Integrals in Bonnet's Form. Let f(x) = e^{-x}cos x, a = 0 and b = p.By the Mean Value Theorem for Integrals in Bonnet's Form, there exists c ∈ (0, p) such that\int_0^p e^{-x}cos x dx = f(c) (p - 0) = e^{-c}cos c (p - 0) = sin p Therefore, there exists c ∈ (0, p) such that \int_0^p e^{-x}cos x dx = sin p