Integration of xe^x cosx

To integrate the expression xe^x cosx, we can use the method of integration by parts. This method involves applying the product rule of differentiation in reverse to obtain an integral.

The product rule states that the derivative of the product of two functions u(x) and v(x) is given by:

(uv)' = u'v + uv'

Applying this rule in reverse, we can rewrite the integral of xe^x cosx as:

∫ xe^x cosx dx = ∫ u dv

where u = x and dv = e^x cosx dx. Now, we need to find du and v to proceed with the integration.

Finding du:
To find du, we differentiate u with respect to x:
du/dx = 1

Finding v:
To find v, we integrate dv:
∫ e^x cosx dx

Integrating e^x cosx requires another integration technique known as integration by parts. This technique involves selecting u and dv in a way that simplifies the integral.

Let's choose u = cosx and dv = e^x dx. Now, we need to find du and v:

Finding du:
To find du, we differentiate u with respect to x:
du/dx = -sinx

Finding v:
To find v, we integrate dv:
∫ e^x dx = e^x

Now, we have all the necessary components to apply integration by parts:

∫ xe^x cosx dx = ∫ u dv
= uv - ∫ v du
= x(e^x) - ∫ e^x (-sinx) dx

Simplifying further, we get:

∫ xe^x cosx dx = x(e^x) + ∫ e^x sinx dx

Now, we have two integrals to evaluate: ∫ e^x sinx dx and ∫ e^x cosx dx. These can be solved using integration by parts or other appropriate techniques.

By following these steps, we can integrate the expression xe^x cosx and obtain the final result.