Petrus
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Hello MHB,
I got problem to antiderivate $$\frac{8x^2}{x^2+2}$$
Regards,
I got problem to antiderivate $$\frac{8x^2}{x^2+2}$$
Regards,
I get $$8- \frac{16}{x^2+2}$$ it should be correct.Ackbach said:I would probably go for polynomial long division here. What do you get?
Petrus said:I get $$8- \frac{16}{x^2+2}$$ it should be correct.
I only get .. $$8x-\frac{8}{x}\ln(x^2+2)$$ and that is wrongAckbach said:That's what I get, too. And you can check it by simply getting a common denominator and adding back up. So where do you go from here?
Petrus said:I only get .. $$\frac{8}{x}\ln(x^2+2)$$ and that is wrong
$$8x-\frac{16}{\sqrt{2}}\arctan(\frac{x}{\sqrt{2}})-16C$$ is that correct?Ackbach said:You have to integrate term-by-term now. You have
$$\int \left( 8- \frac{16}{x^{2}+2} \right)dx
=\int 8 \, dx-16 \int \frac{dx}{x^{2}+2}.$$
Surely you can do the first one. Can you recognize what the second one is?
Petrus said:$$8x-\frac{16}{\sqrt{2}}\arctan(\frac{x}{\sqrt{2}})+C$$ is that correct?
Petrus said:Hello.
I get same answer when I derivate so it's correct! This just show that I should train more antiderivate/derivate! Better early then later!:)
Regards,
Hello Ackbach,Ackbach said:The more correct words in English are that you "get the same answer when you differentiate". Better words are "differentiate" and "anti-differentiate" or better yet, "integrate".
Ackbach said:You have to integrate term-by-term now. You have
$$\int \left( x- \frac{16}{x^{2}+2} \right)dx
=\int x \, dx-16 \int \frac{dx}{x^{2}+2}.$$
Surely you can do the first one. Can you recognize what the second one is?
Prove It said:No, the integral was actually [math]\displaystyle \int{8 - \frac{16}{x^2 + 2}\,dx} [/math].