Integrate with respect to x:(i)12x+3(ii)cos4x+3 - Maths -

Question

I n t e g r a t e   w i t h   r e s p e c t   t o
  x : ( i ) 1 2 x + 3 ( i i ) cos 4 x + 3

Brijendra Pal · Accepted Answer

Hi, we know that ∫cosx dx = sinx +C so second:∫cos4x+3 dxlet 4x+3  = t such that 4dx = dt or dx = dt4∫cost dt4=14∫cost dt = 14 sin t +C = 14 sin 4x+3 +C 
we know that ∫dxx= ln x+Cso here let 2x +3 = t such that 2dx = dt so here  ∫dx2x+3=∫1tdt2=12ln t+C=12ln 2x+3+C

I n t e g r a t e w i t h r e s p e c t t o x : ( i ) 1 2 x + 3 ( i i ) cos 4 x + 3

$I n t e g r a t e w i t h r e s p e c t t o x : (i) \frac{1}{2 x + 3} (i i) \cos (4 x + 3)$