$ \int_{0}^{\infty}e^{x}\cos(x)dx $
I'm trying to solve the integral,
I got: $\int_{0}^{\infty}e^{x}\cos(x)dx=\lim_{b\to\infty}\left(\frac{1}{2}e^{x}(\sin(x)+\cos(x))\right)\mid_{0}^{b}$
but I don't know how to solve this limit, I got:
$$
\lim_{b\to\infty}\left(\frac{1}{2}e^{x}(\sin(x)+\cos(x))\right)\mid_{0}^{b}=\lim_{b\to\infty}\left(\frac{1}{2}e^{b}(\sin(b)+\cos(b))-\frac{1}{2}\right)
$$
but the value of $\sin(b)+\cos(b)$ can be negative/positive…
thanks, and sorry if I have English mistakes…
Best Answer
You are concerned, whether $$ \lim _{b\to\infty} e^{b}(\sin b + \cos b)$$ exists. On the one hand, along the sequence $b_n = \frac{3\pi}{4} + 2n\pi$, the limit is zero. On the other hand, for $b_n = \frac{\pi}{2} + 2n\pi$ the limit is $\infty$. The initial limit does not exist as $b\to\infty$. Consequently, the improper integral diverges.