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
Given $f'(x) = 6x^2 + 2$ and $f(1) = 5$ , find $f(x)$ .
Level 4 · 4 marksTry it
2
A curve passes through $(2, 5)$ and has $\dfrac{\mathrm{d}y}{\mathrm{d}x} = x^3$ . Find $y$ .
Level 4 · 4 marksTry it
3
Given $f'(x) = 3x^2 - 4x + 5$ and $f(0) = -2$ , find $f(2)$ .
Level 4 · 4 marksTry it
4
A curve passes through the point $(1, 7)$ and satisfies

$\dfrac{\mathrm{d}y}{\mathrm{d}x} = 6x^2 - 2x + 3$

Find the equation of the curve.
Level 4 · 4 marksTry it
5
A curve is such that $\dfrac{dy}{dx} = \dfrac{6}{\sqrt{3x+1}}$ for $x > -\dfrac13$ , and the curve passes through the point $(1,\,10)$ .
(a) Find the equation of the curve. (4 marks)
(b) Find the exact area of the region bounded by the curve, the $x$ -axis and the lines $x = 0$ and $x = 5$ . (4 marks)
Level 4 · 8 marksTry it