integration by parts

Integration by Parts is another fundamental technique of integration. While substitution is the reverse of the Chain Rule, Integration by Parts is the reverse of the Product Rule. which state the following

$\frac{d}{dx}[f(x)g(x)]=f(x)g^{\prime }(x)+g(x)f^{\prime }(x).$ It is used when you are integrating the product of two different types of functions (e.g., $x$ multiplied by $e^x$, or $x^2$ multiplied by $\sin (x)$). to see more check differentiation rules

The Formula

The formula allows you to trade a difficult integral for an easier one:

$\int udv=uv-\int vdu$

To use this, you must split your original integral into two parts:

1. $u$: The part you will differentiate.
2. $dv$: The part you will integrate.

How to Choose $u$ (The LIATE Rule)

Success depends entirely on choosing the right $u$. If you pick the wrong one, the integral might get harder! Use the LIATE acronym to prioritize which function to set as $u$ (choose the one that comes first in this list):

1. L — Logarithmic functions ($\ln x,\log x$)
2. I — Inverse Trigonometric functions ($\arcsin x,\arctan x$)
3. A — Algebraic functions ($x,x^2,5x^3$)
4. T — Trigonometric functions ($\sin x,\cos x$)
5. E — Exponential functions ($e^x,3^x$)

Everything else that is left over becomes $dv$.

Step-by-Step Example

Problem: Calculate $\int x\sin (x)dx$

1. Choose $u$ and $dv$

• We have $x$ (Algebraic) and $\sin (x)$ (Trigonometric).
• A comes before T in LIATE, so we choose $u=x$
• The rest is $dv$, so $dv=\sin (x)dx$.

2. Differentiate $u$ and Integrate $dv$

• Differentiate $u$: $\frac{du}{dx}=1⟹du=dx$
• Integrate $dv$: $\int dv=\int \sin (x)dx⟹v=-\cos (x)$

3. Plug into the Formula

Formula: $\int udv=uv-\int vdu$

Substitute our parts:

$\int x\sin (x)dx=(x)(-\cos (x))-\int (-\cos (x))dx$

4. Simplify and Solve

$=-x\cos (x)+\int \cos (x)dx$

Now, the new integral $\int \cos (x)dx$ is easy to solve:

$=-x\cos (x)+\sin (x)+C$