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Math Help - Evaluating Integrals

  1. #1
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    Cool Evaluating Integrals

    Hey. So, I'm having trouble figuring out how to solve this integral. And I would really appreciate it if someone could help me.

    \int_0^{\frac{3\pi}{2}}|sinx|dx

    Okay, so I know I need to split the integral because it is an absolute value. But I'm unsure of what values I need to use to do that. Can I just pick any value I want?

    Also, I've been working on this problem, can someone tell me if I am approaching it correctly?

    g(x)=\int_{tanx}^{x^{2}}\frac{1}{\sqrt{2+t^4}}dt

    Here is what I have thus far:
    \int_{tanx}^{0}\frac{1}{\sqrt{2+t^4}}dt+\int_{0}^{  x^{2}}\frac{1}{\sqrt{2+t^4}}dt

    -\int_{0}^{tanx}\frac{1}{\sqrt{2+t^4}}dt+\int_{0}^{  x^2}\frac{1}{\sqrt{2+t^4}}dt

    g'(x)=\frac{-1}{\sqrt{2+tanx^4}}\frac{d}{dx}(tanx)+\frac{1}{\sq  rt{2+x^6}}\frac{d}{dx}(x^2)

    ..and that's all I have so far, but I'm not really sure what I need to do. Or if I'm even doing it correctly.

    Thanks!!!!
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  2. #2
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    Quote Originally Posted by banshee.beat View Post
    Hey. So, I'm having trouble figuring out how to solve this integral. And I would really appreciate it if someone could help me.

    \int_0^{\frac{3\pi}{2}}|sinx|dx

    Okay, so I know I need to split the integral because it is an absolute value. But I'm unsure of what values I need to use to do that. Can I just pick any value I want?

    Mr F says: Draw the graph first by reflecting in the x-axis the parts of the graph of y = sin x that are below the x-axis. Then it should be quite obvious what the intervals need to be.

    Also, I've been working on this problem, can someone tell me if I am approaching it correctly?

    g(x)=\int_{tanx}^{x^{2}}\frac{1}{\sqrt{2+t^4}}dt

    Here is what I have thus far:
    \int_{tanx}^{0}\frac{1}{\sqrt{2+t^4}}dt+\int_{0}^{  x^{2}}\frac{1}{\sqrt{2+t^4}}dt

    -\int_{0}^{tanx}\frac{1}{\sqrt{2+t^4}}dt+\int_{0}^{  x^2}\frac{1}{\sqrt{2+t^4}}dt

    g'(x)=\frac{-1}{\sqrt{2+tan{\color{red}^4} x}}\, \frac{d}{dx}(tanx)+\frac{1}{\sqrt{2+x^{\color{red}  8}}}\, \frac{d}{dx}(x^2)

    ..and that's all I have so far, but I'm not really sure what I need to do. Or if I'm even doing it correctly. Mr F says: It's almost OK (see the mistakes I corrected in red). Obviously the next step is to calculate the derivatives ....

    Thanks!!!!
    ..
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  3. #3
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    Quote Originally Posted by banshee.beat View Post

    Hey. So, I'm having trouble figuring out how to solve this integral. And I would really appreciate it if someone could help me.

    \int_0^{\frac{3\pi}{2}}|sinx|dx
    \int_{0}^{3\pi /2}{\left| \sin x \right|\,dx}=\int_{0}^{\pi }{\left| \sin x \right|\,dx}+\int_{\pi }^{3\pi /2}{\left| \sin x \right|\,dx}, hence, the integral equals \int_{0}^{\pi }{\sin x\,dx}-\int_{\pi }^{3\pi /2}{\sin x\,dx}.
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