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Thread: integration help

  1. #1
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    integration help

    Need help with these examples:

    a) $\displaystyle \int \frac{\sqrt{1+2lnx}}{x}\, dx$

    b) $\displaystyle \int x^5\sqrt{1-x^3}\, dx$
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  2. #2
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    $\displaystyle

    \int x^5\sqrt{1-x^3}\, dx
    $

    Let $\displaystyle u=1-x^3 \Rightarrow \frac{du}{dx}-3x^2$

    $\displaystyle \int x^5 \sqrt{u} \frac{du}{-3x^2} \Rightarrow -\frac{1}{3}\int x^3 \sqrt{u} ~du$

    $\displaystyle \Rightarrow -\frac{1}{3} \int (1-u)\sqrt{u}~du$

    $\displaystyle =-\frac{1}{3} \int \sqrt{u}-u^{\frac{3}{2}}~du$

    $\displaystyle =-\frac{1}{3} \left( \frac{2}{3}u^{\frac{3}{2}}-\frac{2}{5} u^{\frac{5}{2}} \right)+C$

    $\displaystyle =-\frac{2}{3} \left( \frac{1}{3}u^{\frac{3}{2}}-\frac{1}{5} u^{\frac{5}{2}} \right)+C$
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  3. #3
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    $\displaystyle

    \int \frac{\sqrt{1+2lnx}}{x}\, dx
    $

    Let $\displaystyle u=1+2ln \ x \Rightarrow \frac{du}{dx}=\frac{2}{x}$

    $\displaystyle \int \frac{\sqrt{u}}{x}~\frac{x~du}{2}$

    $\displaystyle \frac{1}{2}\int \sqrt{u}~du $

    $\displaystyle \frac{1}{2}.\frac{2}{3} u^{\frac{3}{2}}+C$

    $\displaystyle \frac{1}{3} u^{\frac{3}{2}}+C$
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  4. #4
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    Quote Originally Posted by Bernice View Post
    Need help with these examples:

    a) $\displaystyle \int \frac{\sqrt{1+2lnx}}{x}\, dx$

    b) $\displaystyle \int x^5\sqrt{1-x^3}\, dx$
    a) Substitute $\displaystyle u = 1 + 2 \ln x$.

    b) Substitute $\displaystyle u = 1 - x^3$.
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  5. #5
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    a more direct substitution is to replace those $\displaystyle u$ by $\displaystyle u^2.$
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