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Evaluate the following integral $$\int\left(\operatorname{cos}\frac{x}{2}\cdot x^3+\frac12\right)\sqrt{4-x^2}\:\:dx$$

My work: I thought if doing it by integration by parts. So I evaluated some integrals individually. $$\int\sqrt{4-x^2}dx=\dfrac{x\sqrt{4-x^2}}{2}+2\arcsin\left(\dfrac{x}{2}\right)+c$$ and $$\int \left(\operatorname{cos}\frac{x}{2}\cdot x^3+\frac12\right)dx=16\left(\sin\left(\dfrac{x}{2}\right)\left(\dfrac{x^3}{8}-3x\right)+\cos\left(\dfrac{x}{2}\right)\left(\dfrac{3x^2}{4}-6\right)\right)+\dfrac{x}{2}+c$$ What to do now, I'm stuck. Any help is greatly appreciated.

IMO the answer is $$\dfrac{x\sqrt{4-x^2}+4\arcsin\left(\frac{x}{2}\right)}{4}+c$$ Can anybody review$?$

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    Is this the famous wi-fi integral meme (without the bounds)? – Zacky Aug 06 '22 at 12:35
  • The original question had bounds $-2,2$ –  Aug 06 '22 at 12:35
  • @Zacky what is wi fi integral –  Aug 06 '22 at 12:36
  • you must use integration by parts to eliminate the factor $ x^3 $ – Jose Garcia Aug 06 '22 at 12:37
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    Wi-Fi integral. Also you will definitely need some weird special functions perhaps to solve the indefinite integral, so better keep the bounds. – Zacky Aug 06 '22 at 12:38
  • With symmetric bounds you can remove most of it by symmetry. I don't think an elementary antiderivative exists. – Klaus Aug 06 '22 at 12:40
  • The bounds, IMO, are actually important as in, they help us to remove the trigonometric functions ($x^3\cos x/2$) part. I could replace that with any crazy odd function like $$\frac {x^2(\arctan x)^2(x-\sin x)}{\cos^{2020}x\ln|x|}$$ and still keep the same answer for the definite integral. – insipidintegrator Aug 06 '22 at 12:41

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This is the famous wifi integral

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