# Thread: Intergration help, reverse chain rule, product rule

1. ## Intergration help, reverse chain rule, product rule

Hi, could someone please verify that this is indeed the correct method for this integral:

$\displaystyle \int x(x^2+8)^2 = \frac{x}{2x.3} (x^2+8)^3 = \frac{1}{6} (x^2+8)^3 + C$.

If there was a constant outside the brackets then I could have done this no problem, just wasn't too sure what to do with the x outside the brackets.

Thanks for the help.

2. It looks correct. You can always differentiate the antiderivative to see if you get the correct answer.

3. Or you could simply multiply it out: $\displaystyle (x^2+ 8)^2= x^4+ 16x^2+ 64$ so $\displaystyle x(x^2+ 8)^2= x^5+ 16x^3+ 64x$ and the integral is $\displaystyle \frac{1}{6}x^6+ 4x^4+ 32x^2+ C$.

$\displaystyle \frac{1}{6}(x^2+ 8)^3+ C= \frac{1}{6}x^6+ \frac{3(8)}{6}x^4+ \frac{3(64)}{6}x^2+ 512+ C$$\displaystyle = \frac{1}{6}x^6+ 4x^4+ 32x^2+ 512+ C$ which only differs from the previous result by the constant of integration.

4. Thanks for the replys, I should have thought of just differentiating the integral!

Was trying to avoid multiplying out, hoping that my method was right

Thanks again