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: -$\dfrac{(2 - x)^9}{9} + c$