Finding the Anti-Derivative of x*cosh(x^2) using Hyperbolic Identities

• Gondur
In summary, the problem is finding the antiderivative of \int xcosh(x^2) dx using the by parts formula and hyperbolic identities. However, u-substitution can be used to make the problem simpler, with u = x^2 and du = 2x dx. This leads to the integral becoming (1/2)∫cosh(u)du, which can be easily solved.
Gondur

Homework Statement

Find the anti derivative of $$\int xcosh (x^2) dx$$

Homework Equations

By parts formula and Hyperbolic Identities of sinh x and cosh x as well as others

The Attempt at a Solution

$$\int xcosh (x^2) dx$$

The problem I'm having is integrating $$\int cosh (x^2) dx$$

I tried setting variables $$u=x$$ and $$\frac{dv}{dx}= \int cosh (x^2) dx$$ with the assumption this could be solved using the by parts formula.

I then concentrated specifically on solving $$\int cosh (x^2) dx$$. I haven't found a method that I know of that's appropriate given that the composite is (x^2) and not (cosh x)^2. Wolfram Alpha shows the solution with an error function - which I know nothing about yet.

I've touched up on Euler's formula $$cosx+isinx=e^{ix}$$ and its parallel $$sinhx+coshx=e^x$$ and I'm just about to learn its applications, maybe it should be used here. This area is new to me so light explanations are wise at this time.

You don't need parts, all you need is u-substitution. u = x^2 and du = 2x dx

Panphobia said:
You don't need parts, all you need is u-substitution. u = x^2 and du = 2x dx

I tried that but gave up because of the extraneous x which would mean substituting it for $$\sqrt {u}$$.

The x in the numerator cancels ut the out in x in the denominator.

Sorry I got it.

look
u = x^2
du = 2x * dx

du/2 = x * dx

(1/2)∫cosh(u)du

Now from there its pretty easy as you can see.

Panphobia said:
look
u = x^2
du = 2x * dx

du/2 = x * dx

(1/2)∫cosh(u)du

Now from there its pretty easy as you can see.

Yes the x variables cancel each other out. I figured it out

1. What is hyperbolic integration?

Hyperbolic integration is a mathematical technique used to solve integrals involving hyperbolic functions, such as sinh(x) and cosh(x). It is a branch of calculus that is closely related to traditional integration methods, but with a focus on hyperbolic functions.

2. How does hyperbolic integration differ from traditional integration?

Hyperbolic integration involves the use of hyperbolic functions, which are a type of trigonometric functions. These functions have different properties and behave differently than traditional trigonometric functions, so the techniques used to integrate them are also different.

3. What are some common applications of hyperbolic integration?

Hyperbolic integration is commonly used in physics and engineering, particularly in fields such as electromagnetics, fluid dynamics, and heat transfer. It is also used in mathematical models of real-world phenomena, such as population growth and radioactive decay.

4. Is hyperbolic integration difficult to learn?

Like any mathematical technique, hyperbolic integration can be challenging to learn at first. However, with practice and a solid understanding of traditional integration methods, it can become easier to grasp. It is important to have a strong foundation in calculus before attempting to learn hyperbolic integration.

5. Are there any special tools or software required for hyperbolic integration?

Hyperbolic integration can be done by hand using traditional integration techniques, such as integration by parts and substitution. However, there are also software programs and online calculators available that can assist with more complex integrals involving hyperbolic functions.

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