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Math Help - Please check this integral for me

  1. #1
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    Please check this integral for me

    Hi can sum1 check this trig integral for me please, thanks.

    evaluate the integral of

    ( cot(x) + csc(x) ) / sin(x) dx

    I got:

    csc(x) (cot(x) + csc(x)) dx

    expanding

    (csc(x)cot(x)) +csc^2(x) dx

    int csc^2(x) dx + int csc(x)cot(x) dx

    int csc^2(x) dx + -csc(x)

    -cot(x) - csc(x) + C
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  2. #2
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    Quote Originally Posted by shadow85 View Post
    Hi can sum1 check this trig integral for me please, thanks.

    evaluate the integral of

    ( cot(x) + csc(x) ) / sin(x) dx

    I got:

    csc(x) (cot(x) + csc(x)) dx

    expanding

    (csc(x)cot(x)) +csc^2(x) dx

    int csc^2(x) dx + int csc(x)cot(x) dx

    int csc^2(x) dx + -csc(x)

    -cot(x) - csc(x) + C
    Try taking the derivative of your answer.
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  3. #3
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    \displaystyle \frac{\cot{x} + \csc{x}}{\sin{x}} = \frac{\frac{\cos{x}}{\sin{x}} + \frac{1}{\sin{x}}}{\sin{x}}

    \displaystyle = \frac{\frac{\cos{x} + 1}{\sin{x}}}{\sin{x}}

    \displaystyle = \frac{\cos{x} + 1}{\sin^2{x}}

    \displaystyle = \frac{\cos{x}}{\sin^2{x}} + \frac{1}{\sin^2{x}}

    \displaystyle = \frac{\cos{x}}{\sin^2{x}} + \frac{\cos^2{x}}{\cos^2{x}\sin^2{x}}

    \displaystyle = \frac{\cos{x}}{\sin^2{x}} + \frac{\sec^2{x}}{\tan^2{x}}.


    So \displaystyle \int{\frac{\cot{x} + \csc{x}}{\sin{x}}\,dx} = \int{\frac{\cos{x}}{\sin^2{x}}\,dx} + \int{\frac{\sec^2{x}}{\tan^2{x}}\,dx}.

    Make the substitution \displaystyle u = \sin{x} so that \displaystyle du = \cos{x}\,dx and make the substitution \displaystyle v = \tan{x} so that \displaystyle dv = \sec^2{x}\,dx and the integrals become

    \displaystyle \int{u^{-2}\,du} + \int{v^{-2}\,dv}

    \displaystyle = \frac{u^{-1}}{-1} + \frac{v^{-1}}{-1} + C

    \displaystyle = -\frac{1}{\sin{x}} - \frac{1}{\tan{x}} + C

    \displaystyle = -\csc{x} - \cot{x} + C.
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    thank you.
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