Question

integrate (cos8x+1)dx/(cot2x-tan2x)

Answer 1

the answer must be like..Acos8x

Answer 2

i got an answer but it was not in this form and was too lengthy to be converted to this form..

Answer 3

@ParthKohli @pooja195 @zepdrix @Jaynator495 @mathstudent55 @mathmale

Answer 4

@imqwerty @UnkleRhaukus @freckles

Answer 5

can we use that ?
I mean can we solve it like this:
\[\text{ assume } \int\limits \frac{\cos(8x)+1}{\cot(2x)-\tan(2x)} dx=A \cos(8x)+C \\ \\ \text{ differentiate both sides } \\ \frac{\cos(8x)+1}{\cot(2x)-\tan(2x)}=-8A \sin(8x) \\ \text{ insert } x=\frac{\pi}{3} \\ \frac{\cos(\frac{8}{3}\pi)+1}{\cot(\frac{2}{3}\pi)-\tan(\frac{2}{3}\pi)}=-8A \sin(\frac{8}{3}\pi) \\ \frac{\frac{-1}{2}+1}{\frac{-1}{\sqrt{3}}+\sqrt{3}}=-8 A \frac{\sqrt{3}}{2} \\ \frac{\frac{1}{2}}{\frac{-1}{\sqrt{3}}+\sqrt{3}}=-4 A \sqrt{3} \\ \frac{-1}{4 \sqrt{3}} \cdot \frac{\frac{1}{2}}{\frac{-1}{\sqrt{3}}+\sqrt{3}}=A \\ \frac{-1}{4} \cdot \frac{\frac{1}{2}}{-1+3} =A \\ A=\frac{-1}{8} \cdot\frac{1}{2}=\frac{-1}{16}\]
Are you allowed to do it this way?

Answer 6

or are you actually trying to put in a pretty form that we can integrate

Answer 7

If you wanted it that way you can write everything in terms of cos and sin
and use double angle identity and half angle identity a lot

Answer 8

\[\cot(2x)-\tan(2x)=\frac{\cos(2x)}{\sin(2x)}-\frac{\sin(2x)}{\cos(2x)} \\ =\frac{\cos^2(2x)-\sin^2(2x)}{\sin(2x) \cos(2x)} \\ = \frac{\cos(4x)}{\frac{1}{2} \sin(4x)} \\ = 2 \frac{\cos(4x)}{\sin(4x)} \\ \text{ so you have } \\ \frac{\cos(8x)+1}{\cot(2x)-\tan(2x)}=\frac{\cos(8x)+1}{2 \cdot \frac{\cos(4x)}{\sin(4x)}} =\frac{\cos(8x)+1}{2} \cdot \frac{\sin(4x)}{\cos(4x)} \\ =\cos^2(4x) \frac{\sin(4x)}{\cos(4x)}=\cos(4x) \sin(4x)\]
and then use a double angle identity and you have an easy form to integrate

...

Answer 9

and that form will lead the the answer above where A is known of course

Answer 10

thank you @freckles i understood pretty well..it was an objective q..so i could do it either way but i wanted to learn the proper method..even i simplified using double angle identity..but at last i got a long answer which contained inverse..so converting to cos8x would be lengthy ..! Thanks again!