Integration By Parts Formula: How To Integrate The Product Of Two Functions In Calculus.

Integration by parts formula

∫udv= uv-∫vdu

The integration by parts formula is a technique used in calculus to integrate the product of two functions. It is denoted by the following formula:

∫u dv = uv − ∫v du

Here, u and v are functions of the variable being integrated with respect to, and du and dv are their respective differentials.

To use this formula, one typically chooses u and dv in a strategic way such that the integral of v du is easier to compute than the original integral. This is often done by picking u to be a function that becomes simpler when differentiated, and dv to be a function that becomes simpler when integrated.

For example, consider integrating the function f(x) = x cos(x). To do so using integration by parts, one can set u = x and dv = cos(x) dx, since the differential of u, du, is simply dx, and the integral of dv can be computed easily as sin(x). Applying the integration by parts formula, we get:

∫x cos(x) dx = x sin(x) – ∫sin(x) dx

which simplifies to:

∫x cos(x) dx = x sin(x) + cos(x) + C,

where C is the constant of integration.

This method can be used to solve a wide range of integration problems, including trigonometric functions, exponential functions, and logarithmic functions.

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