U Substitution Formula

U-substitution is also known as integration by substitution in calculus, u-substitution 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 u-substitution 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 u-substitution 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 u-substitution 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 integrated 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.

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: -\(\dfrac{(2 - x)^9}{9} + c\)