Math Problem Statement

Use a change of variables or the accompanying table to evaluate the integral: ∫(5/3) dx / (9x^2 + 25) from 0 to 5. Write the integral in terms of u.

Solution

This problem involves a substitution method (change of variables) to rewrite the integral in terms of uu. Here's how to solve it step by step:

The given integral is:

0539x2+25dx\int_0^5 \frac{3}{9x^2 + 25} \, dx

Step 1: Recognize the denominator form

The denominator 9x2+259x^2 + 25 can be rewritten as (3x)2+52(3x)^2 + 5^2, resembling the standard form of the trigonometric substitution a2+u2a^2 + u^2.

Step 2: Define the substitution

Let: u=3xso thatdu=3dxanddx=13du.u = 3x \quad \text{so that} \quad du = 3 \, dx \quad \text{and} \quad dx = \frac{1}{3} du.

Step 3: Adjust the limits of integration

When x=0x = 0: u=3(0)=0.u = 3(0) = 0. When x=5x = 5: u=3(5)=15.u = 3(5) = 15.

Step 4: Rewrite the integral

Substitute u=3xu = 3x into the integral: 0539x2+25dx=0153u2+2513du.\int_0^5 \frac{3}{9x^2 + 25} \, dx = \int_0^{15} \frac{3}{u^2 + 25} \cdot \frac{1}{3} \, du.

Simplify: 0539x2+25dx=0151u2+25du.\int_0^5 \frac{3}{9x^2 + 25} \, dx = \int_0^{15} \frac{1}{u^2 + 25} \, du.

Final Answer:

The integral in terms of uu is: 0151u2+25du.\int_0^{15} \frac{1}{u^2 + 25} \, du.

Would you like me to proceed with evaluating the integral or explain more about the substitution process? Let me know!


Here are 5 related questions for further exploration:

  1. What is the integral of 1u2+a2du\int \frac{1}{u^2 + a^2} \, du, and how does it relate to arctangents?
  2. How do you perform substitutions when the limits of integration are not given explicitly?
  3. Can this integral be solved using trigonometric identities instead of substitution?
  4. How does the substitution u=axu = ax generalize for other forms of integrals?
  5. What is the geometric interpretation of the substitution method in integrals?

Tip: Always confirm that the derivative of your substitution aligns perfectly with the original integral to avoid errors!

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Math Problem Analysis

Mathematical Concepts

Integration
Change of Variables
Trigonometric Substitution

Formulas

Substitution: u = g(x), dx = du / g'(x)
Integral of 1 / (u^2 + a^2): (1/a) * arctan(u/a) + C

Theorems

Trigonometric Substitution Theorem
Fundamental Theorem of Calculus

Suitable Grade Level

Grades 11-12 or introductory college-level calculus