Math Problem Statement
integrate (sin(sqrt(x))/(sqrt(x) * cos^3(sqrt(x)))) dx
Solution
We are tasked with evaluating the integral:
Step 1: Substitution
We begin by making the substitution , which implies that: Also, note that , so the integral becomes:
This simplifies to:
Step 2: Use Trigonometric Identity
We now focus on the integral: We recognize that this can be solved by using a simple substitution. Since the derivative of is , let’s make the substitution:
Thus, the integral becomes:
Step 3: Solve the New Integral
The integral of is straightforward:
Thus, the integral becomes:
Step 4: Back Substitution
Since , we substitute back to get:
Now, multiplying by the factor of 2 we had from earlier:
Step 5: Final Answer
Recalling that , we substitute back to get the final answer:
Thus, the integral is:
Would you like further clarification on any of these steps?
Here are 5 related questions to explore:
- What is the integral of ?
- How does trigonometric substitution work in integrals involving sine and cosine?
- How do you integrate functions involving square roots in the integrand?
- What are some alternative methods for integrating trigonometric functions?
- How can you recognize when to use substitution in integral problems?
Tip: When you see trigonometric functions with powers, think about identities or substitution to simplify the integration.
Ask a new question for Free
By Image
Drop file here or Click Here to upload
Math Problem Analysis
Mathematical Concepts
Trigonometry
Integration
Substitution
Formulas
Substitution: u = sqrt(x), dx = 2u du
Integral of v^(-3): -1/2v^2
Theorems
Trigonometric identities
Substitution rule in integrals
Suitable Grade Level
Grade 12 or Higher (Advanced Calculus)