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Thread: Integral with parameters

  1. #1
    Senior Member DeMath's Avatar
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    Integral with parameters

    Calculate

    $\displaystyle \int\limits_0^{+\infty}e^{-ax^2} \cos(bx^2)\,dx,~~~a>0$
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  2. #2
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    Quote Originally Posted by DeMath View Post
    Calculate

    $\displaystyle \int\limits_0^{+\infty}e^{-ax^2} \cos(bx^2)\,dx,~~~a>0$
    starting from $\displaystyle \int_0^{\infty} e^{-x^2} \ dx = \frac{\sqrt{\pi}}{2}$ and letting $\displaystyle x^2=st,$ where $\displaystyle s$ is considered to be the variable, we'll get $\displaystyle \frac{1}{\sqrt{t}}=\frac{1}{\sqrt{\pi}} \int_0^{\infty} \frac{1}{\sqrt{s}}e^{-st} \ ds. \ \ \ \ \ (1)$

    next let $\displaystyle J=\int_0^{\infty} \frac{x+a}{\sqrt{x}[(x+a)^2 + b^2]} \ dx, \ \ a >0.$ let $\displaystyle x=z^2$ to get $\displaystyle J=2\int_0^{\infty} \frac{z^2 + a}{(z^2+a)^2 + b^2} \ dz.$ let $\displaystyle u=z - \frac{\sqrt{a^2+b^2}}{z}, \ v=z+\frac{\sqrt{a^2+b^2}}{z}.$ then:

    $\displaystyle 2\int \frac{z^2 + a}{(z^2+a)^2 + b^2} \ dz=\left(1+\frac{a}{\sqrt{a^2+b^2}} \right)J_1 + \left (1 - \frac{a}{\sqrt{a^2+b^2}} \right)J_2,$ where $\displaystyle J_1=\int \frac{du}{u^2 + 2\sqrt{a^2+b^2}+2a}$ and $\displaystyle J_2=\int \frac{dv}{v^2 - (2\sqrt{a^2+b^2} - 2a)}.$

    both $\displaystyle J_1,J_2$ are easy to find. we'll eventually get: $\displaystyle J=\pi \sqrt{\frac{\sqrt{a^2+b^2} + a}{2(a^2+b^2)}}. \ \ \ \ \ \ \ \ \ (2)$

    now let $\displaystyle I=\int_0^{\infty} e^{-ax^2} \cos(bx^2) \ dx, \ \ a>0.$ put $\displaystyle x=\sqrt{t}$ to get $\displaystyle I=\frac{1}{2} \int_0^{\infty} \frac{e^{-at} \cos(bt)}{\sqrt{t}} \ dt = \frac{1}{2\sqrt{\pi}} \int_0^{\infty} e^{-at} \cos(bt) \int_0^{\infty} \frac{1}{\sqrt{s}}e^{-st} \ ds \ dt,$ by (1).

    changing the order of integration will give us $\displaystyle I=\frac{1}{2\sqrt{\pi}} \int_0^{\infty} \frac{1}{\sqrt{s}} \int_0^{\infty} e^{-(s+a)t} \cos(bt) \ dt \ ds=\frac{1}{2\sqrt{\pi}} \int_0^{\infty} \frac{s+a}{\sqrt{s}[(s+a)^2 + b^2]} \ ds,$ and therefore by (2):

    $\displaystyle I=\sqrt{\frac{\pi(\sqrt{a^2+b^2} + a)}{8(a^2+b^2)}}. \ \ \Box$

    the final answer makes sense to me. so i can claim that i have survived from all the annoying algebra involved in there.
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  3. #3
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    NonCommAlg :

    Do you have any integrals which involve just a little algebra but our deep thinking ,

    In fact , I cannot survive from the annoying annoying algebra !!
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  4. #4
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    Quote Originally Posted by simplependulum View Post
    NonCommAlg :

    Do you have any integrals which involve just a little algebra but our deep thinking ,
    i guess i've already posted many of them?
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