How to find the integral of x(3x^2 + 5)^1/4

  • Thread starter brutalmadness
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    Integral
In summary, the attempt at a solution for this problem was to use the power rule for integrals (x^n+1/n+1). However, the problem gets sloppy at some point and the student gets stuck.
  • #1
brutalmadness
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Homework Statement


[tex]\int[/tex]x(3x^2+5)^1/4

2. The attempt at a solution
u=3x^2+5 du=6x
1/6[tex]\int[/tex]6x(3x^2+5)^1/4

this is where I'm getting sloppy, i think...

1/6[tex]\int[/tex]du(u)^1/4
1/6[tex]\int[/tex]4/5(u)^5/4
[tex]\int[/tex]2/15(3x^2+5)^5/4

and then i get stuck.
 
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  • #2
brutalmadness said:

Homework Statement


[tex]\int[/tex]x(3x^2+5)^1/4

2. The attempt at a solution
u=3x^2+5 du=6x
1/6[tex]\int[/tex]6x(3x^2+5)^1/4

this is where I'm getting sloppy, i think...

1/6[tex]\int[/tex]du(u)^1/4

Good so far...(except du=6xdx and you forgot to include a few dummy variables throughout the problem, but you are accounting for them in your attempt--just remember to write them down too)

brutalmadness said:
1/6[tex]\int[/tex]4/5(u)^5/4
[tex]\int[/tex]2/15(3x^2+5)^5/4

Where are you getting the integral sign from in this part of the problem?
 
  • #3
i was trying to use the power rule for integrals (x^n+1/n+1). but i have a feeling i can't do that. haha :)

then i pulled the 1/6 back into it and multiplied it out (1/6 times 4/5=2/15).
 
  • #4
No, you were doing it right, no need to second guess yourself. The power rule for integrals is all you need to solve this.

Try to integrate the following again:

1/6[tex]\int[/tex]u^(1/4)du

Rather, another way for me to put it:

is [tex]\int[/tex]udu = [tex]\int[/tex]0.5u^2du ?
 
Last edited:
  • #5
no, they wouldn't be equal. for example, does 5=0.5(5)^2? no; 5 does not equal 12.5

did i pull my denominator up incorrectly? if (x^n+1/n+1) is (x^.25+1/.25+1), then (x^1.25/1.25). so to pull up the denominator...
 
  • #6
Ok, if those two aren't equal, then why would

1/6[tex]\int[/tex]du(u)^1/4 = 1/6[tex]\int[/tex]4/5(u)^5/4

Your denominator and everything is fine, you just have an extra integral sign floating around after you already correctly evaluated the integral. I didn't outright say it since I thought you'd notice it, but I suppose my references were a bit too vague.

Once you get rid of the extra integral sign at the point where you are "stuck", you would have 2/15(3x^2+5)^5/4 (+C), and that would simply be your final answer, unless there was an initial condition or something.
 
  • #7
oh wow, that's what i get for trying to do calculus late at night. i was trying to evaluate an integral after i already evaluated it. so my final answer was correct... just need to watch out and make sure i drop the integral sign.

should have been...
(1/6)(4/5)u^5/4
2/15(3x^2+5)^5/4+C

thank you so much.
 
Last edited:

1. What is the general process for finding the integral of an expression?

The general process for finding the integral of an expression is to first identify the variable in the expression and then use the appropriate integration techniques, such as substitution or integration by parts, to solve the integral.

2. How do I determine the limits of integration for a given integral?

The limits of integration can be determined by considering the boundaries of the region being integrated over. This can be done by looking at the graph of the function or by setting up the integral based on the given problem.

3. What is the power rule for integration and how is it applied?

The power rule for integration states that for an expression of the form x^n, the integral is (x^(n+1))/(n+1) + C, where C is the constant of integration. This rule can be applied by using it to integrate each term of the expression separately.

4. Can the integral of x(3x^2 + 5)^1/4 be solved using a substitution?

Yes, the integral of x(3x^2 + 5)^1/4 can be solved using the substitution u = 3x^2 + 5. This will result in the integral being transformed into a simpler form that can be integrated using the power rule.

5. Is there a way to check my answer for the integral of x(3x^2 + 5)^1/4?

Yes, you can check your answer by taking the derivative of your result and seeing if it matches the original expression. You can also use online integration calculators or graphing software to visualize the integral and compare it to the original function.

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