U Substitution Formula
Usubstitution is also known as integration by substitution in calculus, usubstitution formula is a method for finding integrals. The fundamental theorem of calculus generally used for finding an antiderivative. Due to this reason, integration by substitution is an important method in mathematics. The usubstitution formula is another method for the chain rule of differentiation. This u substitution formula is similarly related to the chain rule for differentiation. In the usubstitution formula, the given function is replaced by 'u' and then u is integrated according to the fundamental integration formula. After integration, we resubstitute the actual function in place of u. Let us learn more about the usubstitution formula in the upcoming sections.
What Is U Substitution Formula?
In the U substitution formula, the main function is replaced by 'u' and then the variable u is integrate according to the fundamental integration formula but after integration we resubstitute the actual function in place of u. U substitution formula can be given as :
\( \int f\left(g\left(x\right)\right){g}’\left(x\right)dx=\int f\left(u\right)du\)
where,
 u = g(x)
 du = \({g}’\left(x\right)dx\)
Let us see how to use the u substitution formula in the following solved examples section.
Solved Examples Using U Substitution Formula

Example 1: Integrate \( \int (2x+6)(x^2+6x)^6dx\) using u substitution formula.
Solution:
Let u = \(x^2+6x\)
So that, du = (2x+6)dx.
Substitute the value of u and du in \( \int (2x+6)(x^2+6x)^6dx\), replacing all forms of x, getting
Using U Substitution Formula,
\( \int (2x+6)(x^2+6x)^6dx = \int (x^2+6x)^6 (2x+6)dx\)
\( \int u^6 du\)
= \(\dfrac{u^7}{7}+c\)
= \(\dfrac{(x^2+6x)^7}{7}+c\)
Answer: \(\dfrac{(x^2+6x)^7}{7}+c\).

Example 2 : Integrate \( \int (2  x)^8 dx\)
Solution:
Let u = (2  x)
So that, du = (1)dx.
Substitute the value of u and du in \( \int (2  x)^8 dx\), replacing all forms of x, getting
Using U substitution formula,
\( \int (2  x)^8 dx = \int u^8 (1)du\)
= \( \int u^8 du\)
=  \(\dfrac{u^9}{9}+c\)
= \(\dfrac{(2  x)^9}{9} + c\)
Answer: \(\dfrac{(2  x)^9}{9} + c\)