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Integral of cos(x+pi/8)^2 dx
The solution
The answer (Indefinite)
[src]
/ 3/x pi\ /x pi\ 4/x pi\ 2/x pi\
| 2*tan |- + --| 2*tan|- + --| x*tan |- + --| 2*x*tan |- + --|
| 2/ pi\ x \2 16/ \2 16/ \2 16/ \2 16/
| cos |x + --| dx = C + ----------------------------------- - ----------------------------------- + ----------------------------------- + ----------------------------------- + -----------------------------------
| \ 8 / 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\ 4/x pi\ 2/x pi\
| 2 + 2*tan |- + --| + 4*tan |- + --| 2 + 2*tan |- + --| + 4*tan |- + --| 2 + 2*tan |- + --| + 4*tan |- + --| 2 + 2*tan |- + --| + 4*tan |- + --| 2 + 2*tan |- + --| + 4*tan |- + --|
/ \2 16/ \2 16/ \2 16/ \2 16/ \2 16/ \2 16/ \2 16/ \2 16/ \2 16/ \2 16/
$$\int \cos^{2}{\left(x + \frac{\pi}{8} \right)}\, dx = C + \frac{x \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{16} \right)}}{2 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{16} \right)} + 4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{16} \right)} + 2} + \frac{2 x \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{16} \right)}}{2 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{16} \right)} + 4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{16} \right)} + 2} + \frac{x}{2 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{16} \right)} + 4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{16} \right)} + 2} - \frac{2 \tan^{3}{\left(\frac{x}{2} + \frac{\pi}{16} \right)}}{2 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{16} \right)} + 4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{16} \right)} + 2} + \frac{2 \tan{\left(\frac{x}{2} + \frac{\pi}{16} \right)}}{2 \tan^{4}{\left(\frac{x}{2} + \frac{\pi}{16} \right)} + 4 \tan^{2}{\left(\frac{x}{2} + \frac{\pi}{16} \right)} + 2}$$
/ ___\ / ___\
|1 \/ 2 | |1 \/ 2 |
pi*|- - -----| pi*|- + -----|
\2 4 / \2 4 /
-------------- + --------------
8 8
$$\frac{\pi \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)}{8} + \frac{\pi \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)}{8}$$
=
/ ___\ / ___\
|1 \/ 2 | |1 \/ 2 |
pi*|- - -----| pi*|- + -----|
\2 4 / \2 4 /
-------------- + --------------
8 8
$$\frac{\pi \left(\frac{1}{2} - \frac{\sqrt{2}}{4}\right)}{8} + \frac{\pi \left(\frac{\sqrt{2}}{4} + \frac{1}{2}\right)}{8}$$
pi*(1/2 - sqrt(2)/4)/8 + pi*(1/2 + sqrt(2)/4)/8
Use the examples entering the upper and lower limits of integration.