Integration by parts is a technique in calculus used to integrate a product of two functions. It involves rewriting the integral of the product of two functions as an expression involving only one of the functions, and its derivative, and the other function and its antiderivative.

The formula for integration by parts is:

∫u dv = uv − ∫v du

where u and v are two functions, and du/dx and dv/dx are their derivatives with respect to x.

To use this formula, we choose one function to be u and the other to be dv. Then, we differentiate u and integrate dv.

Here is an example:

∫x sin x dx

In this example, we choose u = x and dv = sin x dx.

Then, we have du/dx = 1 and v = -cos x.

Applying the formula, we get:

∫x sin x dx = -x cos x + ∫cos x dx

= -x cos x + sin x + C

where C is the constant of integration.

Therefore, the integral of x sin x dx is equal to -x cos x + sin x + C.
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