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

[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.

 

[Panphobia]

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

 

[Gondur]

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.

 

[Panphobia]

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.

 

[Gondur]

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