# Evaluate the definite integral

• sapiental
In summary, the conversation discusses a difficulty with solving a definite integral involving sin(x). The individual is seeking help and is told that the function is odd and the integration region is even, which can simplify the solution. The conversation ends with a confirmation and further explanation of the odd function property.
sapiental
hi,

I've been having difficulty with this integral for some time now and any help would be gratly appreciated.

$$\int\frac{x^2 \sin x}{1+x^6}dx$$

this is a definite integral from -pi/2 to pi/2

The sinx has been giving me problems because if I set u = to any part of the equation I can't write sin(u)

for example

u = x^2 du/2 = xdx
u = 1+x^6 du/6 = x^5dx
u = 1+x^3 du/2 = x^2dx

in all these cases I still get stuck with the sinx..

hints on how to approach this equation would be ideal becasue I need to learn how to do this myself.

Thank you!

You won't be able to get the indefinite integral for that thing. But fortunately, you can use the fact the function is odd and the integration region is even.

hey thanks for the help everyone.

StatusX, what you're saying is that I can just write the final result as

$$\int \frac{x^2 \sin x}{1+x^6}dx = 0$$

and the function is odd because of sin(x) right?thanks

Last edited:
Yea, because it is an even function times an odd function.

## 1. What is a definite integral?

A definite integral is a mathematical concept that represents the area under the curve of a function between two given points on the x-axis. It is denoted by ∫(f(x))dx, where f(x) is the function and dx represents an infinitely small change in x.

## 2. How is a definite integral evaluated?

A definite integral is evaluated by finding the antiderivative of the function, plugging in the upper and lower limits of integration, and subtracting the results. This can be done through various methods such as the Fundamental Theorem of Calculus, integration by parts, and substitution.

## 3. What is the difference between a definite and indefinite integral?

A definite integral has specific limits of integration and represents a single numeric value, while an indefinite integral has no limits and represents a family of functions. An indefinite integral is also known as an antiderivative.

## 4. What are some common applications of definite integrals?

Definite integrals are used in many areas of science and engineering, such as physics, economics, and statistics, to calculate quantities such as area, volume, work, and probability. They are also used to solve differential equations and model real-world processes.

## 5. How can I check if my evaluated definite integral is correct?

You can check your answer by taking the derivative of the result. If the derivative matches the original function, then your evaluated definite integral is correct. You can also use online calculators or graphing software to graph the original function and the definite integral to visually confirm the result.

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