Evaluate the definite integral

In summary, the definite integral from a = (-pi/2) to (pi/2) of f(((x^2)(sinx))/(1+x^6))dx can be solved using the substitution u = x and du = dx. However, this substitution does not change the overall integral. A better approach is using the substitution x^{3} = \tan \theta, x = \sqrt[3]{\tan \theta} to simplify the integral and make it easier to solve.
  • #1
sapiental
118
0
the definite integral:

from a = (-pi/2) to (pi/2) f(((x^2)(sinx))/(1+x^6))dx

this is the way it seems most logic to me to set it up using substitution:

u = x
du = dx

from a = (-pi/2) to (pi/2) f(((u^2)(sinu))/(1+u^6))du

= (((-cos(u))(1/3u^3))/(u+1/7u^7))+CI know how to evaluate it from here, I just need some feedback on my substitution setup.

Thanks in advance.
 
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  • #2
sapiental said:
u = x
du = dx
That sort of substitution does nothing but change the letter denoting the variable.
 
  • #3
Try sketching the graph. Notice anything?

courtrigrad said:
Take the square root of the numerator and denominator and use the substitution [tex] x^{3} = \tan \theta, x = \sqrt[3]{\tan \theta} [/tex] You end up getting [tex] \frac{1}{3} \int_{-\frac{\pi}{2}}^{\frac{\pi}{2}} \tan \theta [/tex]

I'm worried about what "Take the square root of the numerator and denominator" could possibly mean? I don't see how your substitution gives what you claim it gives either.
 
  • #4
yeah, I made a mistake. I took the integral of the square root of the function instead of the actual function.
 

1. What is a definite integral?

A definite integral is a mathematical concept used to find the area under a curve between two specific points on a graph. It is represented by the symbol ∫ and is commonly used in calculus and other branches of mathematics.

2. How do you evaluate a definite integral?

To evaluate a definite integral, you need to follow a specific set of steps. First, you need to identify the limits of integration, which are the two points between which you want to find the area under the curve. Then, you need to find the antiderivative of the function within the integral. Finally, you plug in the limits of integration into the antiderivative and subtract the value at the lower limit from the value at the upper limit.

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

A definite integral has specified limits of integration, whereas an indefinite integral does not. This means that a definite integral gives a specific numerical value as the result, while an indefinite integral gives a function as the result.

4. What are some real-world applications of definite integrals?

Definite integrals have many real-world applications, such as calculating the total distance traveled by an object given its velocity over time, finding the total amount of work done by a variable force, and determining the total mass of an object given its density function. They are also used in economics, physics, and engineering.

5. Are there any special rules for evaluating definite integrals?

Yes, there are several special rules for evaluating definite integrals, such as the power rule, the constant multiple rule, and the sum and difference rule. These rules can help simplify the process of evaluating integrals and make them more manageable.

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