Simplified Integration with the U-Substitution Rule: Step-by-Step Guide and Example

U substitution rule

The u-substitution rule, also known as the method of substitution, is a technique used in calculus to simplify the process of integrating functions

The u-substitution rule, also known as the method of substitution, is a technique used in calculus to simplify the process of integrating functions. It involves making a substitution of variables, replacing an expression in the integrand with a new variable, to manipulate the integral into a form that is easier to integrate.

The u-substitution rule follows the following steps:

Step 1: Choose a substitution. Look for a sub-expression within the integrand that can be simplified by substitution. This sub-expression, represented as u, should be a function of the variable of integration.

Step 2: Find du/dx. Differentiate the expression u with respect to x and express it in terms of dx. This will allow us to express dx in terms of du.

Step 3: Substitute the variables. Replace the sub-expression in the integrand with u and dx with du, based on the relationship established in the previous step.

Step 4: Simplify the integrand. Express the entire integrand in terms of u, which should lead to a simpler expression than the original expression.

Step 5: Evaluate the integral. Integrate the simplified expression with respect to u, treating u as the variable of integration.

Step 6: Convert the variable back. Once the integral is evaluated, convert the variable back to the original variable by substituting u with the original expression that you chose in Step 1.

Let’s take an example to understand how the u-substitution rule is applied:

Example:
Evaluate the integral ∫(2x + 1)^3 dx using the u-substitution rule.

Step 1: Choose a substitution. Let’s choose u = 2x + 1.

Step 2: Find du/dx. Differentiating u = 2x + 1 with respect to x, we get du/dx = 2.

Step 3: Substitute the variables. Substitute u = 2x + 1 and dx = du/2 into the integral:

∫(2x + 1)^3 dx = ∫u^3 * (du/2)

Step 4: Simplify the integrand. Express the entire integrand in terms of u:

∫u^3 * (du/2) = (1/2)∫u^3 du

Step 5: Evaluate the integral. Integrate the simplified expression with respect to u:

(1/2)∫u^3 du = (1/2) * (u^4/4) + C

Step 6: Convert the variable back. Convert the variable back to the original variable by substituting u with 2x + 1:

(1/2) * (u^4/4) + C = (1/2) * ((2x + 1)^4/4) + C

Therefore, the evaluated integral is (1/8) * (2x + 1)^4 + C.

That’s how the u-substitution rule can be applied to simplify the integration of functions. Remember that u-substitution can be a very powerful tool when solving integrals, but it may not always work for every integral. Practice and familiarity with various substitution techniques will help to identify opportunities for using the u-substitution rule effectively.

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