Why Does the Integral of 2xCos(Pi*x^2)dx Not Simplify Using Standard Methods?

• Chaz706
In summary, the general integral form for a trigonometric function works when the variable inside is of first degree, but it fails when the variable goes beyond the first degree. This is due to the Fundamental Theory of Calculus. To solve the integral of 2xCos(Pi*x^2)dx, substitution or integration by parts can be used. Substitution can work by letting u = \pi{x^2} and making the appropriate substitution. To format integrals, one can use special coding techniques.
Chaz706
The General integral for a trig form works whenever the variable inside goes to the first degree.
Example: Sin(x)

But the general integral form for when the variable inside goes beyond the first degree doesn't work.
Example: Sin(x^2), Cos(x^3)

I end up getting an integral whose derivative isn't the original function that I integrated. According to the Fundamental Theory of Calculus, these algorithms can't be correct in these cases.

So how should I solve the integral of 2xCos(Pi*x^2)dx ? I've thought about two things: Substitution and Integration by Parts. Substitution could work, but I get hung up on how to get du. Parts I've tried, but I'm hung up on how to integrate that ugly cosine. Is there another method? Does substitution work? If it does, what's the du? Does Parts work? and how would it work if it does?

Reason why I'm asking: this is one large assignment, and my brain's in knots already from the rest of it.
Furthermore: this is my first post. How do you get that cool coded stuff that makes your integrals look like... integrals?

Ok, I think we'll do substitution.

let $$u = \pi{x^2}$$
$$du = 2\pi{x}dx$$

$$\frac{du}{\pi} = 2xdx$$

Now make the substitution:

$$\int 2x\cos{\pi{x^2}}dx = \frac{1}{\pi}\int\cos{u}du$$

I think you can take it from here.

Jameson

Thanks

Thanks for your help Jameson. And Older Dan too.

1. What is the general form of an integral?

The general form of an integral is ∫f(x)dx, where f(x) represents the integrand and dx represents the infinitesimal change in the variable of integration.

2. How do I solve an integral using the general form?

To solve an integral using the general form, you must first identify the integrand and the variable of integration. Then, you can use various integration techniques such as substitution, integration by parts, or trigonometric substitution to evaluate the integral.

3. Can the general form of an integral be applied to all types of integrals?

Yes, the general form of an integral can be applied to all types of integrals, including definite and indefinite integrals. However, the specific integration technique used may vary depending on the complexity of the integrand.

4. Why is the general form of an integral important?

The general form of an integral is important because it represents a fundamental concept in calculus and is used to calculate various quantities such as areas, volumes, and work. It is also the basis for more advanced integration techniques and applications in fields such as physics and engineering.

5. Are there any limitations to the general form of an integral?

While the general form of an integral can be applied to a wide range of integrals, there are some limitations. For example, it cannot be used to solve improper integrals or integrals with discontinuous or undefined functions. In these cases, alternative methods such as limits or complex analysis may be necessary.

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