Integrate powers of x and standard functions as the reverse of differentiation, including the constant of integration, and integrate linear composites

What this dot point is asking

SEAB wants you to integrate, which is to reverse differentiation: given a derivative, find the function it came from. You must reverse the power rule, remember the constant of integration, know the standard integrals, and handle a function of a linear expression such as $(2x + 1)^4$.

The answer

Integration reverses differentiation

If differentiating $F(x)$ gives $f(x)$, then integrating $f(x)$ recovers $F(x)$. The notation is:

The reverse power rule

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

This is the exact reverse of the differentiation power rule. The case $n = -1$ is special and gives a logarithm.

The constant of integration

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.

Integrating a linear composite

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

The extra division by $a$ undoes the chain-rule factor that differentiation would have produced. This rule works only because the inside is linear; for a non-linear inside the simple division does not apply.

Checking by differentiating

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.

Finding a function from its gradient

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.

Examples in context

Example 1. Recovering position from velocity. Given a particle's velocity as a function of time, integrating it returns the displacement, with the constant of integration fixed by a known starting position, which is how kinematics rebuilds motion.

Example 2. Total from a rate. If water flows into a tank at a known rate per second, integrating that rate over time gives the total volume added, the accumulation idea that integration captures.

Try this

Q1. Find $\displaystyle\int x^3\,dx$. [1 mark]

• Cue. $\dfrac{x^4}{4} + C$.

Q2. Find $\displaystyle\int (5x - 2)^3\,dx$. [3 marks]

• Cue. $\dfrac{(5x - 2)^4}{5 \times 4} + C = \dfrac{(5x - 2)^4}{20} + C$.

Q3. Find $\displaystyle\int \left(2x + \dfrac{1}{x^2}\right) dx$. [3 marks]

• Cue. $x^2 - \dfrac{1}{x} + C$.