Math Problem Statement

Evaluate the integral \( \int x \sqrt{5 + 2x^2} \, dx \).

Solution

The given integral is:

x5+2x2dx\int x \sqrt{5 + 2x^2} \, dx

Solution:

To solve this, we use substitution. Let:

u=5+2x2u = 5 + 2x^2

Then:

dudx=4x    du=4xdx\frac{du}{dx} = 4x \quad \implies \quad du = 4x \, dx

From this, we have:

xdx=14dux \, dx = \frac{1}{4} du

Substituting into the integral:

x5+2x2dx=u14du\int x \sqrt{5 + 2x^2} \, dx = \int \sqrt{u} \cdot \frac{1}{4} \, du

Simplify:

x5+2x2dx=14u1/2du\int x \sqrt{5 + 2x^2} \, dx = \frac{1}{4} \int u^{1/2} \, du

Integral of u1/2u^{1/2}:

u1/2du=23u3/2\int u^{1/2} \, du = \frac{2}{3} u^{3/2}

So:

14u1/2du=1423u3/2=16u3/2\frac{1}{4} \int u^{1/2} \, du = \frac{1}{4} \cdot \frac{2}{3} u^{3/2} = \frac{1}{6} u^{3/2}

Substitute u=5+2x2u = 5 + 2x^2 back:

x5+2x2dx=16(5+2x2)3/2+C\int x \sqrt{5 + 2x^2} \, dx = \frac{1}{6} (5 + 2x^2)^{3/2} + C

Final Answer:

x5+2x2dx=16(5+2x2)3/2+C\int x \sqrt{5 + 2x^2} \, dx = \frac{1}{6} (5 + 2x^2)^{3/2} + C


Do you have any questions or need further clarification?

Related Questions:

  1. What is the role of substitution in solving integrals like this?
  2. How do you handle more complex substitutions in definite integrals?
  3. Can you demonstrate integration by parts for a similar type of function?
  4. What happens if the function inside the square root is negative (i.e., complex analysis)?
  5. Can this integral be solved using numerical methods instead of substitution?

Tip:

Always verify your substitution by differentiating the final result to ensure it matches the original integrand!

Ask a new question for Free

By Image

Drop file here or Click Here to upload

Math Problem Analysis

Mathematical Concepts

Integration
Substitution Method

Formulas

Substitution: Let \( u = g(x) \), then \( du = g'(x)dx \)
Power rule for integration: \( \int x^n dx = \frac{x^{n+1}}{n+1} + C \) for \( n \neq -1 \)

Theorems

Integration by substitution

Suitable Grade Level

Grades 11-12