# Integration by Parts of Indefinite Integrals Examples 1

Recall that if we have a function containing two parts to which we can split up and assign one of them "$u$" and the other "$dv$, then it thus follows that:

(1)We will now apply this formula in the following examples.

## Example 1

**Evaluate the following integral: $\int x^2 \ln x \: dx$.**

We let $u = \ln x$ since when differentiated, it becomes simpler and more beneficial than $x^2$, and $dv = x^2 \: dx$. It thus follows that $du = 1/x \: dx$ and $v = x^3/3$. Substituting this into the integration by parts formula we obtain that:

(2)## Example 2

**Evaluate the following integral: $\int x \cos {5x} \: dx$.**

We select $u = x$ since when differentiated, it becomes simpler, and $du = \cos {5x} \: dx$. It thus follows that $du = dx$ and $v = \frac{1}{5} \sin {5x}$. Substituting this into the integration by parts formula and we obtain that:

(3)## Example 3

**Evaluate the following integral: $\int x e^{x/2} \: dx$.**

First let $u = x$ since when differentiated it becomes simpler, and $dv = e^{x/2} \: dx$. It thus follows that $du = dx$ and $v = 2e^{x/2}$. Substituting this into the integration by parts formula and we obtain that:

(4)## Example 4

**Evaluate the following integral: $\int x^2 e^x \: dx$.**

We note that $e^x$ remains unchanged when differentiated, so we let $u = x^2$ and $dv = e^x \: dx$. Therefore, $du = 2x \: dx$ and $v = e^x$. Applying the integration by parts formula we obtain that:

(5)We must now apply integration by parts once again for the rightmost integral. Let's choose $u = x$ and $dv = e^x \: dx$. Therefore, $du = dx$ and $v = e^x$. Substituting this back, we get:

(6)