Quick questions on Integration as the reverse of differentiation explained: O-Level A-Maths

To integrate a power, raise the power by one and divide by the new power:

Differentiation destroys any constant term (its derivative is zero), so integration cannot recover it. We add an arbitrary constant $C$ to every indefinite integral to represent all the functions with the same derivative. Omitting $C$ loses a mark.

For a function of $(ax + b)$, integrate as if it were a simple power, then divide by the coefficient $a$ of $x$ inside:

Integration and differentiation are inverses, so you can always check an integral by differentiating the answer: it should return the integrand. This is the quickest way to catch a missing inner-coefficient division or a wrong power, and examiners reward a confident, self-checked result.

Because integration recovers a function from its derivative, a question that gives $\dfrac{dy}{dx}$ together with a point on the curve can be solved by integrating and then using the point to fix the constant $C$. This turns a gradient function back into the actual equation of the curve.

Find $\displaystyle\int x^3\,dx$. [1 mark]

Find $\displaystyle\int (5x - 2)^3\,dx$. [3 marks]

Find $\displaystyle\int \left(2x + \dfrac{1}{x^2}\right) dx$. [3 marks]