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Math Help - Few integrals

  1. #1
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    Few integrals

    \int \frac {sin(2x) - sin(2k)}{sin(x) - sin(k) + cos(x) - cos(k)}dx

    <br />
\int \frac {dx}{\sqrt {cos^3(x + a)sin(x + b)}}

    \int \sqrt {\frac {cos(x - a)}{cos(x + a)}} dx

    \int \frac {cot(x) - 3cot(x)}{3tan(3x) - tan(x)} dx
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  2. #2
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    For the last one:

    \int\frac{-2cot(x)}{3tan(3x)-tan(x)}dx

    Rewrite as \int\frac{4sin^{2}(x)-1}{4sin^{2}(x)}dx

    =\int{dx}-\frac{1}{4}\int{csc^{2}(x)}dx

    =x+\frac{1}{4}cot(x)+C
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  3. #3
    Behold, the power of SARDINES!
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    Quote Originally Posted by mathwizard View Post
    \int \frac {sin(2x) - sin(2k)}{sin(x) - sin(k) + cos(x) - cos(k)}dx
    Using the sum to product identities we get

    \int \frac{2\cos(x+k)\sin(x-k)}{2\cos\left( \frac{x+k}{2}\right)\sin\left( \frac{x-k}{2}\right)-2\sin\left( \frac{x+k}{2}\right)\sin\left( \frac{x-k}{2}\right)}dx=
    \int \frac{\cos(x+k)\sin(x-k)}{\sin\left( \frac{x-k}{2}\right)\left[\cos\left( \frac{x+k}{2}\right)-\sin\left( \frac{x+k}{2}\right)\right]}dx

    Now using the double angle identities on the numerator we get

    \int \frac{\left[\cos^2\left( \frac{x+k}{2}\right) -\sin^2\left( \frac{x+k}{2}\right) \right]2\sin\left(\frac{x-k}{2}\right)\cos\left( \frac{x-k}{2}\right)}{\sin\left( \frac{x-k}{2}\right)\left[\cos\left( \frac{x+k}{2}\right)-\sin\left( \frac{x+k}{2}\right)\right]}dx

    factoring and reducing we get

    \int 2\cos\left( \frac{x+k}{2}\right)\cos\left( \frac{x-k}{2}\right) -2\sin\left( \frac{x+k}{2}\right)\cos\left( \frac{x-k}{2}\right)dx

    Using the product to sum identities we get

    \int (\cos(x)+\cos(k)-(\sin(x)+\sin(k))dx=\sin(x)+x\cos(k)+\cos(x)-x\sin(k)+C
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