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I was attempting to make integrals to practice solving using various methods. I made this integral,$$\int x^3\tan(x^2)dx$$ , and I have been completely stumped. I am not a master at integration, so just because I couldn't solve it using a method doesn't count it out immediately. With that said, I have tried IBP, Feynmann's, numerous substitutions, and series expansion. It would be much appreciated if anyone could give an idea of how to solve this integral problem.

My attempt at the integral $$\int x^3\tan(x^2)dx$$ $$x^2\rightarrow\ x$$ $$\int x\tan(x)dx$$ IBP $$\frac{1}{2}x\ln(\vert(\sec(x)\vert)-\frac{1}{2}\int \ln(\vert \sec(x)\vert)dx$$ the difficulty is from here on, I believe I could do a Werestrass sub, then probably end up with a dilogorithm based function. not 100% sure though.

charlie
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  • Hi! On MathSE questions are written using MathJax. Please consider editing including what's your attempt on this problem! – Turquoise Tilt Nov 25 '24 at 14:46
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    Substitute $x^{2}=t$. Now the final integral will be of the form of $\int t\tan(t)dt$ – Dev Nov 25 '24 at 14:54
  • Why do you think it should have a nice solution? $tan(1)$ is not special. Use numerical integration – Andrei Nov 25 '24 at 15:54
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    substituting t=x^2 does seem to make it look a little nicer, but from there, the integration is not exactly something I've done before. any further steps after the t sub? – charlie Nov 25 '24 at 17:49
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    WolframAlpha is giving an answer in terms of special functions. – whpowell96 Nov 25 '24 at 17:55
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    After the substitution suggested by @Mathematics, go to https://math.stackexchange.com/questions/2442860/evaluate-the-integral-int-x-tanx-mathrmdx?rq=1. – Gonçalo Nov 25 '24 at 19:50

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