Integrate e^x cos(x)dx, how to do this?

1. Sep 24, 2004

hytuoc

Some one plz show me how to do this problem below
Integral of e^x cos(x) dx
how to I integrate that?
thanks

2. Sep 24, 2004

Pyrrhus

Try Integration by parts

$$\int u dv = uv -\int v du$$

try using $$u = e^x$$ and $$dv = \cos (x) dx$$

Last edited: Sep 24, 2004
3. Sep 24, 2004

Hurkyl

Staff Emeritus
Your book will have this example (or one nearly identical to it) perfored step by step.

4. Sep 24, 2004

Phymath

$$u = e^x \ dv = cos(x) dx$$
$$du = e^x dx \ v = sin(x)$$

$$\int e^x cos(x) dx = e^x sin(x) - \int sin(x) e^x dx$$

$$u = e^x \ dv = sin(x) dx$$
$$du = e^x dx \ v = -cos(x) dx$$
$$\int e^x cos(x) dx = e^x sin(x) - (-e^x cos(x) + \int e^x cos(x) dx)$$
$$= e^x sin(x) + e^x cos(x) - \int e^x cos(x) dx$$
$$\int e^x cos(x) dx + \int e^x cos(x) = e^x (sin(x) + cos(x))$$
$$2 \int e^x cos(x) dx = e^x(sin(x) + cos(x))$$
$$\int e^x cos(x) dx = 1/2 e^x (sin(x) + cos(x))$$ there you go intergration by parts follow
$$u v - \int v du$$

Last edited: Sep 24, 2004
5. Sep 24, 2004

hytuoc

thanks so much