Integral of polynomial to some power

In summary, the formula for finding the integral of a polynomial to some power is ∫x^n dx = (x^(n+1))/(n+1) + C, where n is the power and C is the constant of integration. There are special cases for n = -1, n = 0, and n = 1. To solve an integral of a polynomial to some power, you can use the power rule, integration by parts, or substitution. The integral of a polynomial to some power can be used to find the area under the curve, and you can check your solution by taking the derivative or using online tools.
  • #1
Wiemster
72
0
Does anybody know in general how (if) one can perform the integral of a general polynomial to some, not necessarily integer, power? I.e.

[tex]\int \left(\Sigma_{i=0} ^n c_i x^i \right)^a dx [/tex]

with [itex] c_i [/itex] and [itex] a [/itex]arbitrary (real) numbers,

[tex]\int \left(1+x + x^2 + 2x^5 \right)^{1.7} dx [/tex].

Maybe what I'm looking for is some generalization of Newton's binomium?
 
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  • #2
There is no simple way to do that.
 
  • #3
I already thought it would be difficult, if not impossible. Too bad. Thanks anyway.
 
  • #4
[itex]\int\sqrt{\text{fourth degree}}[/itex] = elliptic integral

[itex]\int_0^1 [x(1-x)]^a\,dx[/itex] = Beta function
 

1. What is the formula for finding the integral of a polynomial to some power?

The formula for finding the integral of a polynomial to some power is ∫x^n dx = (x^(n+1))/(n+1) + C, where n is the power and C is the constant of integration.

2. Are there any special cases when finding the integral of a polynomial to some power?

Yes, for n = -1, the integral becomes ln(x) + C. For n = 0, the integral becomes x + C. And for n = 1, the integral becomes (x^2)/2 + C.

3. How do I approach solving an integral of a polynomial to some power?

To solve an integral of a polynomial to some power, you can use the power rule, which states that the integral of x^n is (x^(n+1))/(n+1), along with any special cases for the specific power. You can also use integration by parts or substitution if necessary.

4. Can I use the integral of a polynomial to some power to find the area under the curve?

Yes, the integral of a polynomial to some power can be used to find the area under the curve. This is because the integral represents the sum of infinitely small areas under the curve, which can be used to approximate the total area.

5. Is there a way to check if my solution for the integral of a polynomial to some power is correct?

Yes, you can check your solution by taking the derivative of the integral and seeing if it matches the original polynomial. If it does, then your solution is correct. Additionally, you can use online calculators or graphing tools to verify your solution graphically.

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