i was thinking about using u-substition at first but i dont think that will work so alright this is how i tried to solve it and i get nowhere by doing it:
∫ x / cos^2x dx.. i wrote it as ∫ 1/cos^2x*x dx
u= 1 / cos2x = (cosx)-2
du = -2*(cosx)-3 *2cosxsinx = -4sinx / (cosx)2
v = x2/2
dv= x
u*v - ∫v*du = x2/2cos2x - ∫-2x2sinx/(cosx)2
I have no idea what to do now
What you've done is good so far, but you've made just a small error. You use u-substitution to set:
u=cos2(x)1=(cos(x))−2
Then you try to use the chain rule to differentiate that. Well, let's make another substitution to illustrate what the chain rule's really doing. Let q = cos(x).
dxdu=dqdu⋅dxdq
dqdu=−2q3
This is part where you went wrong. It appears to me as if you accidentally took the derivative of cos(2x) instead of cos(x).
dxdq=−sin(x)
dxdu=−2q3⋅−sin(x)
Re-substitute cos(x) for q:
dxdu=−2cos3(x)⋅−sin(x)=cos3(x)2sin(x)=cos2(x)2tan(x)
Now, using the integration by parts formula:
∫udv=u⋅v−∫vdu
cos2(x)1⋅2x2−∫2x2⋅cos2(x)2tan(x)dx
Oh. Oops. Looks like that actually made the integral more complicated. If you wanted to continue, you
could use integration by parts again, but it would get far worse before it got better. So, this suggests that the tactic you chose probably wasn't the best one. Unfortunately, the nature of integration by parts is such that you often try something and just end up spinning your wheels, unable to progress. That's a sign you should go back to the drawing board and try different substitutions. In this case, the important thing to remember is that reversing the "order" of the substitutions usually results in following a totally different process, and often one is much easier than the other. Try these substitutions and see where you get. It will go much more smoothly:
u=x⟹du=?
dv=cos2(x)1⟹v=?