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Math Help - Integration Coursework

  1. #1
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    Integration Coursework

    Find

    \int (1+tan(x))/(1+cos(x)) dx

    and

    \int (1+tanh(x))/(1+cosh(x)) dx
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  2. #2
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    Smile

    Integration Coursework-untitled.jpg
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  3. #3
    Moo
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    Hello,

    It is pretty straightforward if you use Weierstrass substitution :

    t=\tan^2 \frac x2

    \cos(x)=\frac{1-t^2}{1+t^2}

    \sin(x)=\frac{2t}{1+t^2}

    \tan(x)=\frac{2t}{1-t^2}

    and dx=\frac{2}{1+t^2} ~ dt
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  4. #4
    MHF Contributor Mathstud28's Avatar
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    Quote Originally Posted by smwatson View Post
    Find

    \int (1+tan(x))/(1+cos(x)) dx
    \int\frac{1+\tan(x)}{1+\cos(x)}dx=\int\bigg[\frac{1}{\cos(x)+1}-\frac{\sin(x)}{\cos(x)(\cos(x)+1)}\bigg]dx
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  5. #5
    Math Engineering Student
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    \begin{aligned}<br />
   \int{\frac{1+\tan x}{1+\cos x}\,dx}&=\int{\frac{(1+\tan x)(1-\cos x)}{\sin ^{2}x}\,dx} \\ <br />
 & =\int{(1+\tan x)\left( \csc ^{2}x-\cot x\csc x \right)\,dx} \\ <br />
 & =\int{\csc ^{2}x\,dx}-\int{\cot x\csc x\,dx}+\int{\frac{dx}{\sin x\cos x}}-\int{\csc x\,dx}.<br />
\end{aligned}

    Each one of these are pretty staightforward to make.
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