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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: $\int \frac{(3+ln+2x)^{3}}{x}dx$

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 $\frac{1}{x}$ .

Then:

$du=\frac{1}{x}dx$

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

$=\frac{u^{4}}{4}+k$

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

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