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U Substitution Practice Problems with Solutions
U substitution is a powerful technique in calculus that allows us to simplify integrals by making a substitution for a variable within the integral. This method is particularly useful when dealing with complex integrals that involve functions within functions. In this article, we will explore some practice problems involving U substitution and provide step-by-step solutions to help you master this important calculus concept.
Understanding U Substitution
Before we dive into practice problems, let’s review the basic idea behind U substitution. The key idea is to make a substitution for a variable within the integral in order to simplify the expression. This substitution is typically denoted by the variable “u,” hence the name U substitution.
Practice Problems
Problem 1:
Calculate the integral of ∫(2x + 1)(x^2 + x)^3 dx using U substitution.
Solution:
Let u = x^2 + x.
. Then, du = (2x + 1)dx. Substituting these into the integral, we get:
- ∫u^3 du
- =(1/4)u^4 + C
- =(1/4)(x^2 + x)^4 + C
Problem 2:
Find the integral of ∫e^(2x)cos(2x) dx using U substitution.
Solution:
Let u = sin(2x). Then, du = 2cos(2x)dx. Substituting these into the integral, we get:
- (1/2)∫e^u du
- =(1/2)e^u + C
- =(1/2)e^(sin(2x)) + C
Key Takeaways
U substitution is a valuable tool in calculus that can simplify complex integrals. By making a strategic substitution for a variable within the integral, we can often transform a difficult problem into a more manageable one. Practice problems like the ones discussed in this article can help you build your skills and confidence in using U substitution effectively.
For more practice problems and in-depth explanations of U substitution, check out this Khan Academy article.
