Quick questions on Integration as the reverse of differentiation: N(A)-Level Additional Mathematics

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

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.

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 $\dfrac{x^{3/2}}{3/2} = \dfrac{2}{3}x^{3/2}$.

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.

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.

When a point is supplied, you are expected to find $c$; leaving the answer with $c$ in it loses the final mark.

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

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

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