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How to do step by step solutions for any integral substitute rule question?

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Last updated date: 23rd Feb 2024
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Answer
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Hint: Here, we will use the definition of u-substitution method. Integral Substitute rule or a u-substitution is a process of evaluating the integrals by substituting the variables for derivatives and antiderivatives. Using this we will discuss the steps involved in using the Integral substitute rule with an example. Integration is the process of adding the small parts to find the whole parts.

Formula Used:
We will use the following formulas:
1. Derivative formula: \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\]
2. Integral Formula: \[\int {{x^n}dx} = \dfrac{{{x^{n + 1}}}}{{n + 1}}\]

Complete Step by Step Solution:
We know in the Integral Substitute rule, we substitute the complex function in variables by using the ILATE rule.
If there is an integral function \[g\left( x \right)\], then we substitute the integrand \[g\left( x \right)\] as \[u\] and \[g'\left( x \right)\] which is also an integrand in the integral function.
So if \[g\left( x \right) = u\], then \[du = g'\left( x \right)dx\] .
Now, we will evaluate the integral in terms of \[u\].
Finally, we will rewrite the terms in terms of \[x\]and the expression \[g\left( x \right)\].
Now we will use the integral substitute rule in the following example.
Example: Evaluate: \[\int {6x{{\left( {3{x^2} + 7} \right)}^3}} dx\]
Now, we will substitute the function \[\left( {3{x^2} + 7} \right)\] as \[u\].
\[u = 3{x^2} + 7\]
Now, we will differentiate the variable with respect to \[x\] by using the derivative formula \[\dfrac{d}{{dx}}\left( {{x^n}} \right) = n{x^{n - 1}}\], we get
\[\begin{array}{l} \Rightarrow \dfrac{{du}}{{dx}} = 6x\\ \Rightarrow du = 6xdx\end{array}\]
So, we can write \[\int {6x{{\left( {3{x^2} + 7} \right)}^3}} dx\] as:
\[\int {6x{{\left( {3{x^2} + 7} \right)}^3}} dx = \int {{u^3}du} \]
Now, we will integrate the variable by using the integral formula \[\int {{x^n}dx} = \dfrac{{{x^{n + 1}}}}{{n + 1}}\], we get
\[ \Rightarrow \int {6x{{\left( {3{x^2} + 7} \right)}^3}} dx = \dfrac{{{u^4}}}{4} + c\]
Now, again by rewriting the expression in terms of \[x\] , we get
\[ \Rightarrow \int {6x{{\left( {3{x^2} + 7} \right)}^3}} dx = \dfrac{{{{\left( {3{x^2} + 7} \right)}^4}}}{4} + c\]
Therefore, this would be the process of integrating a function by using the Integral substitute rule.

Note:
We know that while performing the Integration by substitution, we will substitute the integrand if the integrand is a radicand under a root, the base in a power with real exponent, the exponent with a real base, and the denominator in a fraction. Substitution rule is used to determine the antiderivatives. The substitution method is also a method of integrating functions of various variables and is also used in evaluating the integral using Integration by Parts formula. If the first substitution did not work then try it with a different integrand for substitution.