Makematics
Well-Known Member
Re: MX2 Integration Marathon
![](https://latex.codecogs.com/png.latex?\bg_white \\~~=\sec\theta\tan^3\theta-3\int\sec\theta\tan^2\theta~d\theta-3I\\\\4I=\sec\theta\tan^3\theta-3\int\sec\theta(\sec^2\theta-1)d\theta\\\\~~~~=\sec\theta\tan^3\theta+3\int\sec\theta~d\theta-3\int\sec^3\theta~d\theta\\\\~~~~=\sec\theta\tan^3\theta+3\ln{\left | \sec\theta+\tan\theta \right |}-\frac{3}{2}(\ln{\left | \sec\theta+\tan\theta \right |}+\sec\theta\tan\theta)+c\\\\8I=2\sec\theta\tan^3\theta+3\ln{\left | \sec\theta+\tan\theta \right |}-3\sec\theta\tan\theta+c\\\\\therefore I=\frac{1}{4}x^3\sqrt{1+x^2}+\frac{3}{8}\ln{\left | x+\sqrt{1+x^2} \right |}-\frac{3}{8}x\sqrt{1+x^2}+c)
It is likely that I made a mistake but here goes... Also I skipped a few steps (IBP of sec^3theta dtheta) to save some time.