Given f’(x) = x + b, f(1) = 5 and f(2) = 13

On integrating the given equation, we have

We know

Recall and

On substituting x = 1 in f(x), we get

….. (1)

On substituting x = 2 in f(x), we get

⇒ 13 = 2 + 2b + c

⇒ 13 – 2 = 2b + c

⇒ 2b + c = 11 ….. (2)

By subtracting equation (1) from equation (2), we have

On substituting the value of b in equation (1), we get

∴ c = –2

On substituting the values of b and c in f(x), we get

Thus,