(i) We are looking for a function whose derivative is cos2x
We can recall that,
ddxsin2x=2cos2x
⇒cos2x=12ddx(sin2x)
⇒cos2x=ddx(12sin2x)
Hence, the anti-derivative of cos2x is 12sin2x
(ii) We are looking for a function whose derivative is 3x2+4x3
As we know,
ddx(xn)=nxn−1
∴ddx(x3+x4)=3x2+4x3
Hence, the anti-derivative of 3x2+4x3 is x3+x4
(iii) We know that
⇒ddx(logx)=1x,when x>0⋯(i)
and
⇒ddx[log(−x)]=1−x(−1)=1x,when x<0⋯(ii)
Combining above equations, we get
⇒ddx(log|x|)=1x, x≠0
Hence, the anti-derivatives of 1x is (log|x|)