Why doesn't substitution or integration by parts work for this integral?

Question

Why doesn't integration by parts or u-substitution work for this integral?

∫ x^2 * sin(x^2) dx

Answer 1

Hi Angel!

Let me try and walk through this integral with you! I will try and approach this integral with each method and explain where we get stuck...

Part #1: u-Substitution

∫ x² * sin(x²) dx

It would be very useful if we could use u-substitution to simplify the integral, and hopefully simplify the term inside the trig function... so the most logical choice of "u" value is...

u = x²

So, we can replace all the x² terms with just u...

∫ x² * sin(x²) dx

∫ u * sin(u) dx

This seems good at first glance -- this looks like maybe we could use integration by parts to evaluate -- but remember that we also need to convert that "dx" term into a "du" term!

u = x²

du = 2x * dx

dx = ( 1 / 2x ) * du

Okay... now plug that back into our integral so that our differential is now "du" instead of "dx"...

∫ u * sin(u) dx

∫ ( 1 / 2x ) * u * sin(u) du

Here we hit a roadblock. We have failed to convert all our terms from x's into u's and thus we can't integrate this function.

Part #2: Integration-By-Parts

So let's abandon trying to use u-substitution and just try to integrate-by-parts...

∫ x² * sin(x²) dx

Integration-by-parts is a little tricky, but its a method we can use when we are integrating a function that is a product of two terms, which I'll call "u" and "v". The formula for doing so is as follows:

∫ u*v*dx = [ u * ( ∫vdx ) ] - [ ∫ (u')*( ∫vdx )dx]

When choosing which term should be "u" and which should be "v", we want to pick "u" to be a term which gets simpler and simpler each time it is differentiated. In this case, since the 3rd derivative of x² will equal zero, I will define u = x²... Now let's evaluate all the relevant parameters...

u = x²

v = sin( x² )

u' = 2x

∫ v dx = ∫ sin( x² ) dx

Hmm.. How can we handle that ∫ sin( x² ) dx term? We can't use u-substitution, or even integration-by-parts again. This is where we hit our roadblock with this method.

It turns out that the answer for the ∫ sin( x² ) dx integral is what mathematicians call a transcendental function, which is basically a rare case where it is not solvable analytically. So we are really unable to proceed further here, at least in this level of math.

If you instead picked your u and v terms oppositely, then you'd run into a different problem with integration-by-parts, namely, you'd have to do an infinite loop of integration-by-parts on the results of your previous integration-by-parts. You can investigate that result if you so choose.

Hope that helps!

--Zach