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Thread: Trig integrals

  1. #1
    Junior Member symstar's Avatar
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    Trig integrals

    Two problems in this question, the first I was able to do but was getting an incorrect answer and the second I'm just plain stumped on.

    FIRST Q:
    $\displaystyle \int\cot^3{x}\csc^3{x}dx$

    Here is what I've done so far... I'm coming close but my numbers aren't quite right.

    $\displaystyle u=\csc^2{x}$
    $\displaystyle du=-\cot{x}dx$
    $\displaystyle \csc{x}=\sqrt{u}$

    $\displaystyle \int (\csc^2{x}-1)\cot{x}\csc^3{x}dx$
    $\displaystyle -\int (u-1)u\sqrt{u}du$
    $\displaystyle -\int (u^{2}-u)u^{1/2}du$
    $\displaystyle -\int u^{5/2}du + \int u^{3/2}du$
    $\displaystyle \tfrac{2}{5}u^{5/2}-\tfrac{2}{7}u^{7/2}+C$

    ...and subbing in just leaves me with the wrong answer. Where did I go wrong?

    SECOND Q:

    $\displaystyle \int\csc{x}dx$

    Thought to use $\displaystyle \csc{x}=\sqrt{1+\cot^2{x}}$ or $\displaystyle \frac{1}{\sin{x}}$, but neither really panned out

    Thanks in advance.
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  2. #2
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    Quote Originally Posted by symstar View Post
    $\displaystyle u=\csc^2{x}$
    $\displaystyle du=-\cot{x}dx$
    $\displaystyle \csc{x}=\sqrt{u}$


    The derivative of $\displaystyle \csc^2{x}$ is not $\displaystyle -\cot{x}$.

    Pull one factor of $\displaystyle \csc{x}\cot{x}$ and recall that:

    $\displaystyle (\csc{x})' = -\csc{x}\cot{x}$

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

    As for your second question, multiply it with $\displaystyle \frac{\csc{x}+\cot{x}}{\csc{x}+\cot{x}}$ and see where that gets you.
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  3. #3
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    Quote Originally Posted by symstar View Post
    [snip]
    SECOND Q:

    $\displaystyle \int\csc{x}dx$

    Thought to use $\displaystyle \csc{x}=\sqrt{1+\cot^2{x}}$ or $\displaystyle \frac{1}{\sin{x}}$, but neither really panned out

    Thanks in advance.
    The conventional approach is to use the Weierstrass substitution: The Weierstrass Substitution Example
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  4. #4
    Junior Member symstar's Avatar
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    Can't believe I made that mistake... I was thinking of the integral and not derivative. My class isn't up to the Weierstrass substitution yet. Also, I find it funny that a proof for the integral of csc(x) I was viewing stated that the strategy was not obvious... oi!

    Thanks for the help.
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