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Math Help - A quick check on several integral problems,please

  1. #1
    Boz
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    A quick check on several integral problems,please

    These problems are part of my homework. However, I am bit unsure about some of them, so I'd like to ask you whether my answers are correct. If you find mistakes and have the time please write me the full solution. Thx

    Pr.1 \displaystyle \int xe^x dx Let u=x, \frac{du}{dx}=1, du=dx \displaystyle, Let v= \int e^x=e^x<br />
\displaystyle\int udv=uv-\int vdu = xe^x-\int e^x dx=xe^x-e^x=e^x(x-1)

    Pr.2 \displaystyle \int x^2 e^{x^{3}}, Let u=x^2,\frac{du}{dx}=2x, du=2x dx
    Let \displaystylev=\int e^{x^{3}}= e^{x^{2}}/x
    \displaystyle\int u dv= x^2e^{x^{2}}/x- \int 2x{e^{x^{2}}/x=xe^{x^{2}}-2e^{x^{2}}=e^{x^{2}}(x-2)

    Pr.3 \displaystyle\int x^2 e^x dx, Let u=x^2,du/dx=2x, du= 2x dx \displaystyleLet v=\int e^x=e^x Then \int u dv=x^2 e^x -\int e^x2x dx \displaystyle,Let u_{2}=e^x,du_{2}/dx=e^x, du_{2}=e^xdx \displaystyleLet v_{2}=\int 2x=2,\int u_{2}dv_{2}=e^x2-\int 2e^x dx=e^x2-2e^x=0

    \displaystyle\int u dv=x^2e^x

    Pr.4 \displaystyle\int x e^{x^{2}}dx Let \displaystyleu=x,du/dx=1,du=dx, Let v=\int e^{x^{2}}=e^{x^{2}}
    \displaystyle\int u dv=xe^{x^{2}}-\int e^{x^{2}} dx =xe^{x^{2}}-e^{x^{2}}=e^{x^{2}}(x-1)
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  2. #2
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    1 correct

    2 incorrect

    3 not sure what your final answer is here but you don't have it correct

    4 incorrect

    You can always take the derivative of your solutions as a check too.
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  3. #3
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by Boz View Post
    These problems are part of my homework. However, I am bit unsure about some of them, so I'd like to ask you whether my answers are correct. If you find mistakes and have the time please write me the full solution. Thx



    Pr.2 \displaystyle \int x^2 e^{x^{3}}, Let u=x^2,\frac{du}{dx}=2x, du=2x dx
    Let \displaystylev=\int e^{x^{3}}= e^{x^{2}}/x
    \displaystyle\int u dv= x^2e^{x^{2}}/x- \int 2x{e^{x^{2}}/x=xe^{x^{2}}-2e^{x^{2}}=e^{x^{2}}(x-2)



    Pr.4 \displaystyle\int x e^{x^{2}}dx Let \displaystyleu=x,du/dx=1,du=dx, Let v=\int e^{x^{2}}=e^{x^{2}}
    \displaystyle\int u dv=xe^{x^{2}}-\int e^{x^{2}} dx =xe^{x^{2}}-e^{x^{2}}=e^{x^{2}}(x-1)
    For No. 2.. Use the Substitution Rule..

    Let u=x^3 and du=3x^2\;dx \implies dx = \dfrac{du}{3x^2}

    so, \displaystyle \int x^2 e^{x^{3}} = \int x^2\;e^u\;\dfrac{du}{3x^2} = \dfrac{1}{3}\int e^u\;du

    integrate and plug back u..

    Do the same for No. 4. Let u=x^2

    What are you doing for No 3? You need integration by parts!
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  4. #4
    Member integral's Avatar
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    Prb3:
    \displaystyle u=x^2
    \displaystyle v=e^x
    \displaystyle du=2xdx
    \displaystyle dv=e^xdx

    \displaystyle \int udv=uv-\int vdu
    \displaystyle \int x^2 e^x dx=x^2e^x-2\int e^xxdx

    Then int by parts again to get the answer.
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  5. #5
    Boz
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    Quote Originally Posted by integral View Post
    Prb3:
    \displaystyle u=x^2
    \displaystyle v=e^x
    \displaystyle du=2xdx
    \displaystyle dv=e^xdx

    \displaystyle \int udv=uv-\int vdu
    \displaystyle \int x^2 e^x dx=x^2e^x-2\int e^xxdx

    Then int by parts again to get the answer.
    So, the answer would be e^x(x^2-2x+4)??

    And what about:
    pr.2 \frac{e{x^3}}{3}?
    pr.4 \frac{e{x^2}}{2}?
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  6. #6
    Boz
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    Quote Originally Posted by integral View Post
    Prb3:
    \displaystyle u=x^2
    \displaystyle v=e^x
    \displaystyle du=2xdx
    \displaystyle dv=e^xdx

    \displaystyle \int udv=uv-\int vdu
    \displaystyle \int x^2 e^x dx=x^2e^x-2\int e^xxdx

    Then int by parts again to get the answer.
    So, the answer would be e^x(x^2-2x+4)??

    And what about:
    pr.2 \frac{e^{x^3}}{3}?
    pr.4 \frac{e^{x^2}}{2}?
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  7. #7
    MHF Contributor harish21's Avatar
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    Quote Originally Posted by Boz View Post
    So, the answer would be e^x(x^2-2x+4) I think you made a calculation mistake. It should be e^x(x^2-2x+2)??

    And what about:
    pr.2 \frac{e^{x^3}}{3}?
    pr.4 \frac{e^{x^2}}{2}?
    2 and 4 are correct, but dont forget to add the constant.
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  8. #8
    Boz
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    Quote Originally Posted by harish21 View Post
    2 and 4 are correct, but dont forget to add the constant.
    I found the mistake. Thx very much .
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