A-level · Integration

The constant of integrationExam-style questions

Why every indefinite integral carries a + C — differentiation loses the constant, so integrating gives a family of curves. Reading the family as one shape at every height; finding the curve through a given point; starting values in motion, and why C is not always zero; two constants when you integrate twice; and why a definite integral needs no constant.

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9 questions · A-level level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 3 Level 4

1
A curve passes through $(1, 4)$ and has $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 2x$ . Find $y$ in terms of $x$ .
Level 3 · 4 marksTry it
2
A curve passes through $(2, 5)$ and has $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 3x^2 - 2$ . Find $y$ .
Level 3 · 4 marksTry it
3
A curve passes through $(0, 3)$ and has $\dfrac{\mathrm{d}y}{\mathrm{d}x} = 4x - 1$ . Find $y$ .
Level 3 · 4 marksTry it
4
A curve $C$ $C$ has equation $y = \text{f}(x)$ $y = f (x)$ , where

$\text{f}'(x) = 4x^3 - 6x^2 + 3x + k$

and $k$ $k$ is a constant.

The point $Q(2, -6)$ $Q (2, - 6)$
- lies on $C$
- is a turning point of $C$
(a) Show that $k = -14$ $k = - 14$

(2 marks)

The curve $C$ $C$ cuts the $y$ $y$ -axis at the point $R$ $R$ .
(b) Hence find the coordinates of $R$ $R$ .

(3 marks)
Level 3 · 5 marksTry it
5
Given $f'(x) = 6x^2 + 2$ and $f(1) = 5$ , find $f(x)$ .
Level 4 · 4 marksTry it

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