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Thread: More Integrals

  1. #1
    Member FalconPUNCH!'s Avatar
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    More Integrals

    1.$\displaystyle xsin(x^2)cos(3x^2)dx$

    2.
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  2. #2
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    Krizalid's Avatar
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    For the first one, let $\displaystyle u=x^2.$ After turning the resultant product into a sum the rest follows.

    For the second one, let $\displaystyle u=x\ln x.$ The remaining integral can be killed with a trig. or hyperbolic substitution.
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  3. #3
    Member FalconPUNCH!'s Avatar
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    Quote Originally Posted by Krizalid View Post
    For the first one, let $\displaystyle u=x^2.$ After turning the resultant product into a sum the rest follows.

    For the second one, let $\displaystyle u=x\ln x.$ The remaining integral can be killed with a trig. or hyperbolic substitution.
    I got the first one, but I'm still having some trouble with the second one. I'm not really understanding it.
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  4. #4
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    Hello, FalconPUNCH!!

    Krizalid is absolutely correct . . .


    $\displaystyle \int(1+\ln x)\sqrt{1+(x\ln x)^2}\,dx$

    Let $\displaystyle u \:=\:x\ln x\quad\Rightarrow\quad du \:=\:\left(x\!\cdot\!\frac{1}{x} + \ln x\right)\,dx\:=\1 + \ln x)\,dx$


    We have: .$\displaystyle \int\underbrace{\sqrt{1 + (x\ln x)^2}}\underbrace{(1 + \ln x)dx}$
    . . . . . . . . . . . . . $\displaystyle \downarrow\qquad\qquad\swarrow$
    . . . . . . . . $\displaystyle = \;\int \sqrt{1 + u^2}\;du $

    Now let $\displaystyle u \:= \:\tan\theta\quad\Rightarrow\quad du \:=\:\sec^2\!\theta\,d\theta \quad\hdots\quad \text{etc.}$

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