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First of all, many thanks for this great website.

Let $F(x)$ be the antiderivative of $$f(x)=\frac{x^3+\sin(x)}{x^2+2}.$$ If $F(5)=\pi$, then what is $F(2)$?

This question was in Calculus exam, I know that $F(x)=\int f(x)\ \mathrm dx +C$, but I am not sure how to integrate the $f(x)$. I divided the integral into two parts, first one was $\int \frac{x^3}{x^2+2} \ \mathrm dx$ which is easy using the long division , but the second part got me stuck which is $\int \frac {\sin(x)}{x^2+2}\ \mathrm dx$. I tried to use integration by parts but no results at all. I used WolframAlpha to find the answer, which gives a strange answer using complex numbers. I have not studied any complex course so far. So I'm guessing there is another trick here, could anyone help me to find that?

Brian61354270
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F.Turgut
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    I don't think there are any tricks here. You have to use special functions or series to find this value. – bjorn93 Mar 03 '20 at 16:23
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    By Newton-Leibniz $$ F(2) = \pi - \int_2^5 {\frac{{x^3 + \sin x}}{{x^2 + 2}}dx} = - 5.866664349855 \ldots . $$ I do not think there is a nice, closed-form expression for this value. – Gary Mar 03 '20 at 16:43
  • -Gary May you please specify more details about how you got the result? Thanks. – F.Turgut Mar 03 '20 at 17:24
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    @F.Turgut $\int_2^5f(x),dx=F(5)-F(2)$ since $F$ is an antiderivative. You can approximate $F(2)$ numerically if you want. – bjorn93 Mar 03 '20 at 18:12

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