5

I would like to integrate this $$\int x \ln x\sin x\mathrm dx$$

can I do it with another quick method than using integration by part?

  • You'll have to integrate $\frac1x\sin x$ anyways. –  Jun 02 '19 at 02:03
  • 5
    It is not always possible to integrate by more than one way – Tojra Jun 02 '19 at 02:03
  • 1
    Some Magic is helpful in this situation – logo Jun 02 '19 at 02:41
  • 1
    Can you tell us you have a problem with integration by parts? If you want to give up the best method you should have some reason for it. – Kavi Rama Murthy Jun 02 '19 at 04:59
  • 1
    I think this integral can not expressed by the known elementary functions: $$-i/2\ln \left( x \right) {{\rm e}^{ix}}-1/2,\ln \left( x \right) { {\rm e}^{ix}}x+i/2\ln \left( x \right) {{\rm e}^{-ix}}-1/2,\ln \left( x \right) {{\rm e}^{-ix}}x+1/2,\pi,{\it csgn} \left( x \right) -{\it Si} \left( x \right) +i/2{{\rm e}^{-ix}}-i/2{{\rm e}^{i x}} $$ – Dr. Sonnhard Graubner Jun 02 '19 at 05:59
  • 1
    But Mathematica shows that the answer to this integral contains a $Si(x)=\int \frac{\sin x}{x} dx.$ So this is clearly not expected to be doable by any method including the integration by pars! – Z Ahmed Jun 03 '19 at 10:56

1 Answers1

-1

Here is a fun but not very useful trick: if you know the antiderivative already, any integral can be computed using only $u$-substitution. Proof:

$$\int f(x) \, dx$$ Let $F'(x) = f(x)$. $$u = F(x), du = f(x) \, dx$$ $$\int du = u + C = F(x) + C.$$

So for your integral, we see that it can be solved using the magic $u$-substitution

$$u = -Si(x) + \sin x + \ln x (\sin x - x \cos x).$$

We have

$$du = x \ln x \sin x \, dx$$ $$\int du = u + C = -Si(x) + \sin x + \ln x (\sin x - x \cos x),$$

as desired.

Math2718
  • 515
  • Don't entirely understand why this answer was downvoted. If the question is whether or not there was a nicer way to do it than IBP, then the answer is almost certainly that there isn't one. If the question is if IBP is the only method to solve this integral, as I had assumed, then this shows that there is another way. – Math2718 Jun 30 '19 at 21:51