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Thread: plz help! Integration problems

  1. #1
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    plz help! Integration problems

    Alright. I already put a lot of thought into some of these but I cannot get the right answer out or am stuck in the middle.

    1) The integral, from 0 to pi/4, of ((sinx)^4)((cosx)^2)dx

    My thought process has gotten to (1/16)integral(1-cos4x)dx -integral((1/8)((cosx)^2)((sin2x)^2)dx

    What do i do from here? Or is there a different way to go at it?


    2) Integral, from pi/4 to pi/2 of (cotx)^2 for this one, I got all the way to solving the integral, which i Think is -.5(cotx)^2 -ln(abs value of)sinx ...

    I am stuck here, since pi/2 cant go into cotangent... what do I do?

    3) The integral of 1 to radical 3 of arctan(1/x)dx. I solved this several times, and I got the integral to come out as xarctan(1/x) +.5ln(x^2 +1)
    However, this is wrong, because i know that the answer SHOULD be -.468. How do i get -.468?

    THANK YOU! for your helP!
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  2. #2
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    1. I'll have to play with this one ... even powers of sine and cosine are a pain-in-the-...


    2. $\displaystyle \cot^2{x} = \csc^2{x} - 1$

    3. how is it negative ?

    from 1 to sqrt(3), the function y = arctan(1/x) is greater than 0.

    I get positive .468
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  3. #3
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    Hello, 3deltat!

    $\displaystyle 3)\;\;\int^{\sqrt{3}}_1 \arctan\left(\frac{1}{x}\right)\,dx$

    I got the integral to come out as: .$\displaystyle x\arctan\left(\frac{1}{x}\right) + \frac{1}{2}\ln(x^2 +1) $ . . . . Right!

    However, this is wrong, because i know that the answer SHOULD be -0.468 . ??
    How can the answer be negative? . . . The graph is above the x-axis.
    Code:
            |
            |           ..*
            |      .*:::::|
            |   *:::::::::|
            | * |:::::::::|
            |*  |:::::::::|
            |   |:::::::::|
          - * - + - - - - + -
            |   1        √3
    Ah, I see skeeter already beat me to it!


    We have: .$\displaystyle x\arctan\left(\frac{1}{x}\right) + \frac{1}{2}\ln(x^2+1)\:\bigg]^{\sqrt{3}}_1$

    . . $\displaystyle \bigg[\sqrt{3}\arctan\left(\frac{1}{\sqrt{3}}\right) + \frac{1}{2}\ln(4)\bigg] - \bigg[1\!\cdot\!\arctan(1) + \frac{1}{2}\ln(2)\bigg] $

    . . $\displaystyle = \;\bigg[\sqrt{3}\left(\frac{\pi}{6}\right) + \ln\left(4^{\frac{1}{2}}\right)\bigg] - \bigg[\frac{\pi}{4} + \frac{1}{2}\ln(2)\bigg] $

    . . $\displaystyle = \;\frac{\pi\sqrt{3}}{6} + \ln(2) - \frac{\pi}{4} - \frac{1}{2}\ln(2) $

    . . $\displaystyle = \;\left(\frac{2\sqrt{3}-3}{12}\right)\pi + \frac{1}{2}\ln(2) $

    . . $\displaystyle = \;\boxed{0.468}075109$

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  4. #4
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    $\displaystyle \int \sin^4{x}\cos^2{x}$

    Here's my way:

    $\displaystyle \frac{1}{8} \int (1-\cos{2x})^2(1+\cos{2x})$

    $\displaystyle \frac{1}{8} \int (\underbrace{1-\cos^2{2x}}_{\sin^2{2x}})(1-\cos{2x})$

    $\displaystyle \frac{1}{8} \int (\sin^2{2x} - \sin^2{2x}\cos{2x})$

    And we're done...

    Another double angle formula for the first integral and sub $\displaystyle u = \sin{2x}$ for second integral.
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  5. #5
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by 3deltat View Post
    Alright. I already put a lot of thought into some of these but I cannot get the right answer out or am stuck in the middle.

    1) The integral, from 0 to pi/4, of ((sinx)^4)((cosx)^2)dx

    My thought process has gotten to (1/16)integral(1-cos4x)dx -integral((1/8)((cosx)^2)((sin2x)^2)dx

    What do i do from here? Or is there a different way to go at it?
    When both are even, you need to apply these identities: $\displaystyle \sin^2 u=\frac{1-\cos (2u)}{2}$ and $\displaystyle \cos^2u=\frac{1+\cos(2u)}{2}$:

    $\displaystyle \int_0^{\frac{\pi}{4}}\left[\sin^2x\right]^2\cos^2x\,dx=\int_0^{\frac{\pi}{4}}\left[\tfrac{1}{2}(1-\cos (2x))\right]^2\left[\tfrac{1}{2}(1+\cos(2x))\right]\,dx$ $\displaystyle =\tfrac{1}{8}\int_0^{\frac{\pi}{4}}\left[1-\cos (2x)-\cos^2(2x)+\cos^3(2x)\right]\,dx$

    Now split up the integral:

    $\displaystyle \tfrac{1}{8}\int_0^{\frac{\pi}{4}}\left[1-\cos (2x)-\cos^2(2x)+\cos^3(2x)\right]\,dx$ $\displaystyle =\tfrac{1}{8}\int_0^{\frac{\pi}{4}}\left[1-\cos (2x)-\cos^2(2x)\right]\,dx + \tfrac{1}{8}\int_0^{\frac{\pi}{4}}\cos^3(2x)\,dx$

    Let's focus on this integral:

    $\displaystyle \tfrac{1}{8}\int_0^{\frac{\pi}{4}}\left[1-\cos (2x)-\cos^2(2x)\right]\,dx$

    This is the same as saying $\displaystyle \tfrac{1}{8}\int_0^{\frac{\pi}{4}}\left[\tfrac{1}{2}-\cos (2x)-\tfrac{1}{2}\cos(4x)\right]\,dx$

    Evaluating, we get $\displaystyle \tfrac{1}{8}\left.\left[\tfrac{1}{2}x-\frac{1}{2}\sin(2x)-\tfrac{1}{8}\sin(4x)\right]\right|_0^{\frac{\pi}{4}}=\tfrac{1}{8}\left[\tfrac{1}{8}\pi-\tfrac{1}{2}\right]=\frac{\pi-4}{64}$

    Now let's evaluate $\displaystyle \tfrac{1}{8}\int_0^{\frac{\pi}{4}}\cos^3(2x)\,dx$

    Break off a factor of $\displaystyle \cos(2x)$ and apply the identity $\displaystyle 1-\sin^2 u = \cos^2 u$

    Thus, the integral becomes $\displaystyle \tfrac{1}{8}\int_0^{\frac{\pi}{4}}\left[1-\sin^2(2x)\right]\cos(2x)\,dx$

    Now let $\displaystyle z=\sin(2x)\implies \,dz=2\cos(2x)\,dx$

    We can change the limits of integration as well.

    The integral can now be written as $\displaystyle \tfrac{1}{16}\int_0^1\left[1-z^2\right]\,dz=\tfrac{1}{16}\left.\left[z-\tfrac{1}{3}z^3\right]\right|_0^1=\frac{2}{48}$

    Finally, our total solution is $\displaystyle \frac{\pi}{64}-\frac{1}{48}=\color{red}\boxed{\frac{3\pi-4}{192}}$

    Does this make sense?

    --Chris
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  6. #6
    Rhymes with Orange Chris L T521's Avatar
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    Quote Originally Posted by Chop Suey View Post
    $\displaystyle \int \sin^4{x}\cos^2{x}$

    Here's my way:

    $\displaystyle \frac{1}{8} \int (1-\cos{2x})^2(1+\cos{2x})$

    $\displaystyle \frac{1}{8} \int (\underbrace{1-\cos^2{2x}}_{\sin^2{2x}})(1-\cos{2x})$

    $\displaystyle \frac{1}{8} \int (\sin^2{2x} - \sin^2{2x}\cos{2x})$

    And we're done...

    Another double angle formula for the first integral and sub $\displaystyle u = \sin{2x}$ for second integral.
    mmmmmk...

    This is a "bit" easier

    --Chris
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  7. #7
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    Yep this makes more sense! Thank you for your help. =)
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