How to integrate e^x*cos(x)?

To integrate e^x*cos(x), use integration by parts.

Integration by parts is a technique used to integrate the product of two functions. It involves choosing one function to differentiate and the other to integrate. In this case, we will choose e^x as the function to differentiate and cos(x) as the function to integrate.

Let u = cos(x) and dv/dx = e^x. Then, du/dx = -sin(x) and v = e^x.

Using the formula for integration by parts, we have:

∫ e^x*cos(x) dx = e^x*cos(x) - ∫ -e^x*sin(x) dx

= e^x*cos(x) + e^x*sin(x) - ∫ e^x*cos(x) dx

Rearranging, we get:

2∫ e^x*cos(x) dx = e^x(cos(x) + sin(x))

Dividing both sides by 2, we get:

∫ e^x*cos(x) dx = 1/2 * e^x(cos(x) + sin(x)) + C

where C is the constant of integration.

Therefore, the integral of e^x*cos(x) is 1/2 * e^x(cos(x) + sin(x)) + C.

For further exploration of various techniques used in solving integrals, you can review more about integration techniques. To understand the integration of trigonometric functions in detail, visit our notes on integrating trigonometric functions. Additionally, for a broader perspective on both definite and indefinite integrals, see our section on definite and indefinite integrals.

A-Level Maths Tutor Summary: To find the integral of e^x*cos(x), we use the integration by parts technique. By selecting e^x to differentiate and cos(x) to integrate, we apply the formula and rearrange to get the final answer as 1/2 e^x(cos(x) + sin(x)) + C, where C is the constant of integration. This method simplifies the integration of products of functions.