Solved Exercise: Integral Example 5

Solve the indefinite integral:

$$ \int (x+2)^3 \, dx $$

Step-by-Step Solution

We'll solve this integral using the substitution method.

First, set $ u = x + 2 $.

$$ u = x + 2 $$

Then, the differential of $ u = x + 2 $ is:

$$ du = dx $$

Substitute $ u = x + 2 $ and $ dx = du $ into the integral:

$$ \int (x+2)^3 \, dx = \int u^3 \, du $$

This integral is now straightforward to solve:

$$ \int u^3 \, du = \frac{u^4}{4} + c $$

Now, substitute $ u = x + 2 $ back into the solution:

$$ \int u^3 \, du = \frac{(x+2)^4}{4} + c $$

Thus, the solution to the integral is:

$$ \int (x+2)^3 \, dx = \frac{(x+2)^4}{4} + c $$