How to integrate x^2*cos(x)?

To integrate x^2*cos(x), use integration by parts. This technique is essential for handling products of functions as described in our guide on 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. The formula for integration by parts is:

∫u dv = uv - ∫v du

where u and v are functions of x, and du/dx and dv/dx are their respective derivatives.

To integrate x^2*cos(x), let u = x^2 and dv = cos(x) dx. Then, du/dx = 2x and v = sin(x). Substituting these values into the formula, we get:

∫x^2*cos(x) dx = x^2*sin(x) - ∫2x*sin(x) dx

Now, we need to integrate ∫2x*sin(x) dx. Let u = 2x and dv = sin(x) dx. Then, du/dx = 2 and v = -cos(x). Substituting these values into the formula, we get:

∫2x*sin(x) dx = -2x*cos(x) - ∫-2*cos(x) dx

Simplifying this expression, we get:

∫2x*sin(x) dx = -2x*cos(x) + 2*sin(x) + C

where C is the constant of integration.

Substituting this expression back into the original equation, we get:

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

Simplifying this expression, we get:

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

For further learning and a deeper understanding of similar techniques, explore our detailed discussions on techniques of integration and integration of trigonometric functions. These resources offer extensive insights into handling integrals more effectively and cover various methods including integration rules found in basic integration rules.

A-Level Maths Tutor Summary: To find the integral of x^2cos(x), we apply integration by parts twice. First, we choose u = x^2 and dv = cos(x) dx, leading to an intermediate step. Then, integrating 2xsin(x) again by parts gives us the final answer: x^2sin(x) + 2xcos(x) - 2*sin(x) + C, where C is the integration constant.