integration of uv formula

The integration of the UV formula, often referred to as integration by parts, is a technique used in calculus. It is based on the product rule for differentiation and is used to integrate the product of two functions. The formula is expressed as:

$$ \int u \, dv = uv – \int v \, du $$

How to use it

(1) Split the integrand into two parts, $ u $ and $ dv $, such that $ u $ is a function of $ x $ and $ dv $ is a differential of another function of $ x $. The choice of $ u $ and $ dv $ is crucial and can simplify the problem significantly. A common strategy is to choose $ u $ as the part that becomes simpler when differentiated, and $ dv $ as the part that is easy to integrate.
We can decide the u function using the word ILATE
I -> Inverse trigonometric functions
L -> Logarithmic functions
A-> Algebraic functions
T -> trigonometric functions
E -> Exponential functions

(2) Differentiate $ u $ with respect to $ x $ to find $ du $.

(3) Integrate $ dv $ with respect to $ x $ to find $ v $.

(4) Substitute $ u $, $ du $, and $ v $ into the integration by parts formula and simplify.

Examples

Question 1
$\int x \cdot e^x \; dx$

Solution
Let $ u = x $. This is a good choice because its derivative is simpler.
Let $ dv = e^x dx $. This is chosen because the exponential function is easy to integrate.
Then
$ du = dx $.
and
$ v = \int e^x dx = e^x $.

Therefore
$ \int x \cdot e^x dx = x \cdot e^x – \int e^x \cdot dx \\
= x \cdot e^x – e^x + C $

Example 2
$ \int x^2 e^x \, dx $.

Solution
$ u = x^2 $ (since its derivative is simpler).
$ dv = e^x dx $ (easy to integrate).
Then
$ du = 2x dx $.
and
$ v = \int e^x dx = e^x $.

Therefore
$ \int x^2 e^x dx = x^2 e^x – \int 2x e^x dx $.

Now, we need to integrate $ \int 2x e^x dx $ by parts again.

For $ \int 2x e^x dx $:
Choose $ u = 2x $ and $ dv = e^x dx $.
Then, $ du = 2 dx $ and $ v = e^x $.
So, $ \int 2x e^x dx = 2x e^x – \int 2e^x dx = 2x e^x – 2e^x $.

Substituting this back into our original equation:
$ x^2 e^x – (2x e^x – 2e^x) $.

So, the integral is $ x^2 e^x – 2x e^x + 2e^x + C $.