Integrate the following w.r.t. x.
(i) `4x^(3)` , (ii) `x-1/x` , (iii) `1/(2x+3)` , (iv) `cos (4x+3)`
(v) `cos^(2) x`

Explanation

(i) `int 4x^(3) dx=4(x^(3+1)/(3+1))+c=(4x^(4))/(4)+c=x^(4)+c`
(ii) `int(x-1/x)dx=intxdx-int1/xdx=x^(2)/2-ln x+c`
(iii) `int(dx)/(2x+3)=(ln(2x+3))/2+c`
(iv) `int cos(4x+3)dx=(sin(4x+3))/(4)+c`
(v) `intcos^(2) xdx=int(2 cos^(2) x)/(2)dx=int((1+cos 2x))/2dx=1/2intdx+1/2 int cos2xdx`
`=x/2+1/2(sin 2x)/2+c=x/2+1/4 sin 2x+c`