Integrate powers of x by reversing the power rule, including the constant of integration, and integrate simple sums

What this dot point is asking

SEAB wants you to integrate, which means reversing differentiation. Given a derivative, integration recovers the original function (up to a constant). You need the reverse power rule, the constant of integration $c$, and the ability to integrate a sum of terms. Integration is the tool for finding a function from its rate of change and, later, for areas under curves.

The answer

Integration undoes differentiation

If differentiating $F(x)$ gives $f(x)$, then integrating $f(x)$ gives back $F(x)$. The integral sign $\int$ with $dx$ means "find the function whose derivative is this":

The reverse power rule

To integrate a power of $x$, add one to the power and divide by the new power:

This is the exact opposite of the power rule for differentiation, which multiplied by the power and reduced it. A constant multiple stays in front; a constant on its own, say $k$, integrates to $kx$.

The constant of integration

Because differentiating any constant gives zero, integration cannot know what constant was there originally. So every indefinite integral must include $+ c$, the constant of integration. Omitting it is the most common lost mark in this topic.

Integrating a sum

Integrate a sum term by term, just as you differentiate term by term. Rewrite roots and reciprocals as indices first so the power rule applies, for example $\sqrt{x} = x^{1/2}$ integrates to $\dfrac{x^{3/2}}{3/2} = \dfrac{2}{3}x^{3/2}$.

Finding the particular curve

If you are given a point the curve passes through, substitute it after integrating to find the value of $c$. This turns the general family of curves into the one specific curve required.

Checking by differentiating back

Integration and differentiation are exact opposites, which gives you a built-in check: differentiate your answer and you should recover the function you started with. For instance, $\int 6x^2\, dx = 2x^3 + c$, and differentiating $2x^3 + c$ gives $6x^2$ again. Running this quick check at the end catches a mishandled power or a dropped coefficient before it costs marks.

Examples in context

Example 1. Recovering distance from velocity. If velocity is the derivative of displacement, then integrating the velocity recovers the displacement. This is exactly why integration is central to the kinematics topic, where it reverses the differentiation that produced velocity.

Example 2. Total from a rate. If water flows into a tank at a known rate (a function of time), integrating that rate gives the total volume added over a period. Integration turns a rate-of-change function back into a total-quantity function.

Try this

Q1. Find $\displaystyle\int x^4\, dx$. [1 mark]

• Cue. $\dfrac{x^5}{5} + c$.

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

• Cue. $x^2 + 3x + c$.

Q3. Find $\displaystyle\int \left(3x^2 - 4x\right) dx$. [2 marks]

• Cue. $x^3 - 2x^2 + c$.