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Math Help - Some integration questions

  1. #1
    Member roshanhero's Avatar
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    Some integration questions

    1. \int a^xe^xdx
    I tried to solve it by using integration by parts but couldnot get the required answer.
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  2. #2
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    Quote Originally Posted by roshanhero View Post
    1. \int a^xe^xdx
    I tried to solve it by using integration by parts but couldnot get the required answer.
    Try writing a^x = e^{x \ln a}
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    Quote Originally Posted by roshanhero View Post
    1. \int a^xe^xdx
    I tried to solve it by using integration by parts but couldnot get the required answer.
    Assuming a>0
    Actually it must be positive since the base must be positive.
    \int a^{x} e^{x} dx = C1 + a^{x} e^{x} - ln(a) \int a^{x} e^{x} dx
    then
    (1+ln(a)) \int a^{x} e^{x} dx = a^{x} e^{x} + C1
     \int a^{x} e^{x} dx = \frac{a^{x} e^{x}}{1+ln(a)} + C
    where  C = \frac{C1}{1+ln(a)}
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    Quote Originally Posted by roshanhero View Post
    1. \int a^xe^xdx
    I tried to solve it by using integration by parts but couldnot get the required answer.
    Let y = (ae)^x then

    \ln y = x \ln (ae)

    \frac{1}{y} \frac{dy}{dx} = \ln (ae)

    dx = \frac{dy}{y \ln (ae)}

    and

    \int y ~dx = \int y \frac{dy}{y \ln (ae)}

    \int y ~dx = \int \frac{dy}{\ln (ae)}
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  5. #5
    Member roshanhero's Avatar
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    How can we write a^x = e^{x \ln a}
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    Quote Originally Posted by roshanhero View Post
    How can we write a^x = e^{x \ln a}
    see Natural exponents and logarithms
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    Quote Originally Posted by roshanhero View Post
    How can we write a^x = e^{x \ln a}
    Its the definition of a^x .
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  8. #8
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    \int a^xe^xdx=\int e^{xln a}e^x dx<br />
=\int e^{x(log a+1)} dx
    I hope that upto now all is right. Now I can use the formula of integration of e^ax.
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    Quote Originally Posted by roshanhero View Post
    \int a^xe^xdx=\int e^{xln a}e^x dx<br />
=\int e^{x(log a+1)} dx
    I hope that upto now all is right. Now I can use the formula of integration of e^ax.
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  10. #10
    Member roshanhero's Avatar
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    I am sorry that I still cannot prove a^x = e^{x \ln a} even with the link given by you guys.
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  11. #11
    Member roshanhero's Avatar
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    Allright guys I proved it but in a messy way,tell me whether it is right or wrong
    <br />
Let,a^x=y<br />
    <br />
ln a^x=lny<br />
    <br />
a^x=e^{lny}<br />
    <br />
a^x=e^{lna^x}<br />
    <br />
a^x=e^{xlna}<br />
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  12. #12
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    Quote Originally Posted by roshanhero View Post
    Allright guys I proved it but in a messy way,tell me whether it is right or wrong
    <br />
Let,a^x=y<br />
    <br />
ln a^x=lny<br />
    <br />
a^x=e^{lny}<br />
    <br />
a^x=e^{lna^x}<br />
    <br />
a^x=e^{xlna}<br />
    Yeh that's fine, it's probably less messy to just say

    e^{x \ln{a}} = \left( e^{\ln a} \right)^x = a^x
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