Why do we need the constant of integration?

Consider three simple functions F(x)=2x, G(x)=2x-6, H(x)=2x + π/2. We can differentiate these functions with respect to x to get F’(x)=f(x)=2, G’(x)=g(x)=2, H’(x)=h(x)=2. Clearly the derivatives of these functions are all equal to 2, but the functions are not the same (a simple graph would convince us). Going backwards if we’re given u(x)=2 and we are asked to find the antiderivative of u(x) (i.e. a function U(x) such that U’(x)=u(x)) we cannot simply write that U(x)=2x since that would give us only one out of the infinite antiderivatives. Hence, we are looking for a family of functions of the form U(x)=2x+C where C is any real number. To convince ourselves this is a solution we can differentiate with respect to x which gives U’(x)=u(x)=(2x+C)’=(2x)’+(C)’=2*1+0=2.

PC
Answered by Panoraia C. Maths tutor

2850 Views

See similar Maths A Level tutors

Related Maths A Level answers

All answers ▸

A curve has equation y = 20x −x^2 −2x^3 . Find its stationary point(s).


How to differentiate y = xcos(x)


Differentiate y = x^3 + 2x^2 + 4x + 7


Find the equation of the normal to the curve y=2x^3 at the point on the curve where x=2. Write in the form of ax+by=c.


We're here to help

contact us iconContact ustelephone icon+44 (0) 203 773 6020
Facebook logoInstagram logoLinkedIn logo

© MyTutorWeb Ltd 2013–2025

Terms & Conditions|Privacy Policy
Cookie Preferences