 # Integration by Substitution Formula

Integration of substitution is also known as U – Substitution, this method helps in solving the process of integration function.

When a function cannot be integrated directly, then this process is used. To integration by substitution is used in the following steps:

• A new variable is to be chosen, let’s name t “x”
• The value of dx is to is to be determined.
• Substitution is done
• Integral function is to be integrated
• Initial variable x, to be returned.

The standard formula for integration is given as:

$\large \int f(ax+b)dx=\frac{1}{a}\varphi (ax+b)+c$

$\large \int f\left(x^{n}\right)x^{n-1}dx=\frac{1}{n}\phi \left(x^{n}\right)+c$

$\large \int \frac{{f}'(x)}{f(x)}dx=log\:f(x)+c$

Solved Examples

Question: Find the integration using the substitution formula:

$$\begin{array}{l}\int \frac{(3+ln2x)^{3}}{x}dx\end{array}$$

Solution

Let u = 3 + ln 2x
We can expand out the log term on the right hand side as: 3 + ln 2x = 3 + ln 2 + ln x

The first 2 terms on the right are constants (whose derivative equals zero) and the derivative of the natural log of x is

$$\begin{array}{l}\frac{1}{x}\end{array}$$
.

Then:

$$\begin{array}{l}du=\frac{1}{x}dx\end{array}$$

$$\begin{array}{l}\int \frac{(3+ln\; 2x)^{3}}{x}dx= \int u^{3}du\end{array}$$

$$\begin{array}{l}=\frac{u^{4}}{4}+k\end{array}$$

$$\begin{array}{l}=\frac{(3+ln\:2x)^{4}}{4}+k\end{array}$$