U Substitution: Solving Definite Integral of [(x^2sinx)/(1+x^6)]*dx

In summary, U substitution is a technique used in calculus to simplify the process of solving definite integrals by substituting a new variable, typically denoted as u, for a complicated expression within the integral. To determine what to substitute for u, look for a part of the integral that resembles the derivative of the function inside the parentheses. Not all definite integrals can be solved using U substitution, as it is most effective when the integral contains a complicated expression that can be simplified by substituting a new variable. There are a few rules to follow when using U substitution, such as ensuring u is a function of x and substituting u back in for x in the final answer. This technique can be used for both indefinite and definite integrals,
  • #1
jennie312
1
0
The problem: The definite integral of [(x^2sinx)/(1+x^6)]*dx on the interval -∏/2 ≤ x ≤ ∏/2


I need help figuring out what the u should be for substitution.

I've been trying to make the (1+x^6) my u, but I don't know if this is what I should be doing.
 
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  • #2
There's no elementary integral for that. Can you think of some way to show it's zero without doing the integral? Try thinking about what a graph would look like.
 
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FAQ: U Substitution: Solving Definite Integral of [(x^2sinx)/(1+x^6)]*dx

1) What is U substitution and why is it used to solve definite integrals?

U substitution is a technique used in calculus to simplify the process of solving definite integrals. It involves substituting a new variable, typically denoted as u, for a complicated expression within the integral. This helps to simplify the integral and make it easier to solve.

2) How do you determine what to substitute for u in U substitution?

To determine what to substitute for u, look for a part of the integral that resembles the derivative of the function inside the parentheses. For example, in the given integral, [(x^2sinx)/(1+x^6)]*dx, the derivative of (1+x^6) is 6x^5, which is similar to x^2, so we can substitute u = 1+x^6.

3) Can any definite integral be solved using U substitution?

No, not all definite integrals can be solved using U substitution. This technique is most effective when the integral contains a complicated expression that can be simplified by substituting a new variable. If the integral does not contain any such expression, other techniques such as integration by parts may be more suitable.

4) Are there any rules or guidelines to follow when using U substitution?

Yes, there are a few rules to keep in mind when using U substitution. First, the variable u must be a function of x, and the derivative of u must be present in the integral. Additionally, any constants should be kept outside of the integral. Finally, don't forget to substitute u back in for x in the final answer.

5) Can U substitution be used for both indefinite and definite integrals?

Yes, U substitution can be used for both indefinite and definite integrals. In the case of definite integrals, the limits of integration must also be substituted for u. This will result in a new integral with limits in terms of u, which can be solved using the usual techniques.

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