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Math Help - Last math problem for test

  1. #1
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    Last math problem for test

    Here it goes
    ∫ from 0 to 2π/3 tan (x/2) dx

    this is my last problem i need to know for my test on friday so please help me out
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  2. #2
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    start with this integral fact...

    \int \tan{u} \, du = -\ln|\cos{u}| + C
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  3. #3
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    \int_0^{\frac{2\pi}{3}} \tan\left(\frac{x}{2}\right) \, dx

    u = \frac{x}{2}

    du = \frac{1}{2} dx

    substitute and reset the limits of integration ...

    2 \int_0^{\frac{2\pi}{3}} \tan\left(\frac{x}{2}\right) \, \frac{1}{2}dx

    2 \int_0^{\frac{\pi}{3}} \tan(u) \, du

    integrate ...

    2\left[-\ln|\cos{u}|\right]_0^{\frac{\pi}{3}}

    use the FTC ...

    2\left[-\ln\left(\frac{1}{2}\right) + \ln(1)\right] = 2\ln(2)
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