Mastering Integration by Parts: A Step-by-Step Guide for Calculus Students

Integration by parts formula

Integration by parts is a useful technique in calculus that allows us to find the integral of a product of two functions

Integration by parts is a useful technique in calculus that allows us to find the integral of a product of two functions. The formula for integration by parts is derived from the product rule for differentiation.

The formula for integration by parts is given by:

∫ u dv = uv – ∫ v du

Here, u and v are functions of x, and du and dv are differentials of u and v with respect to x, respectively.

In order to apply the integration by parts formula, we need to choose which function to assign as u and which one to assign as dv. A helpful guideline is to choose u as a function that becomes simpler when differentiated, and dv as a function that becomes easier to integrate.

Once we have chosen u and dv, we can find du and v by differentiating and integrating u and dv, respectively. Then, we substitute these values into the integration by parts formula and evaluate the resulting integrals.

Let’s consider an example to illustrate the use of the integration by parts formula:

Example: Find the integral of x * cos(x) dx

Solution:
In this example, we can choose u = x and dv = cos(x) dx.

Differentiating u gives du = dx, and integrating dv gives v = sin(x).

Now, we can substitute these values into the integration by parts formula:

∫ x * cos(x) dx = x * sin(x) – ∫ sin(x) dx

Evaluating the integral on the right side, we obtain:

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

Where C is the constant of integration.

So, the integral of x * cos(x) dx is x * sin(x) + cos(x) + C.

Remember that when applying the integration by parts formula, it may be necessary to repeat the process multiple times if the resulting integral is still difficult to compute.

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