Math Problem Statement

why is answer 0

Why is the answer 0 for the integral \int_{-2\sqrt{5}}^{2\sqrt{5}} \frac{6x}{\sqrt{x^2 + 16}} \, dx?

Solution

The given integral is:

25256xx2+16dx\int_{-2\sqrt{5}}^{2\sqrt{5}} \frac{6x}{\sqrt{x^2 + 16}} \, dx

This is a definite integral of an odd function over a symmetric interval. Let's break this down:

1. The Function Inside the Integral

The function being integrated is:

f(x)=6xx2+16f(x) = \frac{6x}{\sqrt{x^2 + 16}}

Here, f(x)f(x) is an odd function because it satisfies the condition:

f(x)=f(x)f(-x) = -f(x)

This can be checked as follows: f(x)=6(x)(x)2+16=6xx2+16=f(x)f(-x) = \frac{6(-x)}{\sqrt{(-x)^2 + 16}} = \frac{-6x}{\sqrt{x^2 + 16}} = -f(x)

2. Integral of Odd Function Over a Symmetric Interval

When integrating an odd function over a symmetric interval, such as [a,a][-a, a], the result is always 0. This is because the positive and negative parts of the function cancel each other out.

In this case, since the integral is over the symmetric interval [25,25][-2\sqrt{5}, 2\sqrt{5}], and the integrand is an odd function, the result is:

25256xx2+16dx=0\int_{-2\sqrt{5}}^{2\sqrt{5}} \frac{6x}{\sqrt{x^2 + 16}} \, dx = 0

Hence, the answer is 00.

Would you like more details or have any questions?

Related Questions:

  1. What is the general definition of an odd function?
  2. How does symmetry affect definite integrals?
  3. Can an integral of an even function over a symmetric interval be zero?
  4. What would change if the integrand was an even function?
  5. How can you check if a function is odd or even?

Tip:

Always check the symmetry of the integrand when dealing with definite integrals over symmetric intervals—this can help simplify the evaluation!

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Math Problem Analysis

Mathematical Concepts

Definite Integrals
Odd Functions
Symmetry in Calculus

Formulas

\int_{-a}^{a} f(x) dx = 0 \text{ if } f(x) \text{ is odd}
Odd function: f(-x) = -f(x)

Theorems

Odd Function Integral Theorem
Symmetry of Integrals Over Symmetric Intervals

Suitable Grade Level

Grades 11-12, College Calculus