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Math Help - Intergration help, reverse chain rule, product rule

  1. #1
    Super Member craig's Avatar
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    Intergration help, reverse chain rule, product rule

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

    \int x(x^2+8)^2<br />
=<br />
\frac{x}{2x.3} (x^2+8)^3<br />
=<br />
\frac{1}{6} (x^2+8)^3 + C<br />
.

    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.
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  2. #2
    Senior Member Spec's Avatar
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    It looks correct. You can always differentiate the antiderivative to see if you get the correct answer.
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  3. #3
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    Or you could simply multiply it out: (x^2+ 8)^2= x^4+ 16x^2+ 64 so x(x^2+ 8)^2= x^5+ 16x^3+ 64x and the integral is \frac{1}{6}x^6+ 4x^4+ 32x^2+ C.

    \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 = \frac{1}{6}x^6+ 4x^4+ 32x^2+ 512+ C which only differs from the previous result by the constant of integration.
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  4. #4
    Super Member craig's Avatar
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    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
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