A-Level Maths (Edexcel 9MA0) · Pure Year 1 · Chapter 13: Integration · Lesson 1

Integrating xⁿ and finding functions

71 marks

Hints online, answers free with an account:
mathswariz.uk/worksheets/a-level-integrating-x-and-finding-functions/

Date

Do Now

Quick questions to start.

A1
Needed today
Write $\frac{4}{x ^{3}} - 3 x$ in the form $a x^{p} + b x^{q}$ .
[2 marks]
Answer
A2
Last lesson
$V = 48 x - x^{3}$ . Find the positive value of $x$ for which $V$ is stationary.
[2 marks]
Answer $x = \dots\dots$
A3
From chapter 12
Find $\frac{d y}{d x}$ when $y = 6 x^{\frac{4}{3}} + \frac{2}{x}$ .
[2 marks]
Answer
A4
From chapter 1
Expand and simplify $(3 x - 2)^{2}$ .
[2 marks]
Answer

Example, then your turn

A-Level

Your teacher works through each example with you. Then try the one beside it.

Example 1

Find $\int (8 x^{3} - 6 x^{- 4} + 3 x^{\frac{1}{2}}) d x$ .

Hint Add one to each power, then divide by the new power. Do each term on its own, and add

c

once at the end.

B1
Find $\int (10 x^{4} + 9 x^{- 4} - 2 x^{- \frac{1}{2}}) d x$ .
[3 marks]
Answer

Example 2

Find $\int \frac{x ^{2} - 3}{2 x} d x$ .

Hint You cannot integrate the top and the bottom separately. Divide each term on top by

2 x^{\frac{1}{2}}

first.

B2
Find $\int \frac{( 2 x + 3 ) ( x - 4 )}{x} d x$ , giving each term in its simplest form.
[4 marks]
Answer

Example 3

A curve has $\frac{d y}{d x} = 3 x^{2} + k x - 4$ , where $k$ is a constant. The curve has a stationary point where $x = 2$ , and it passes through $(1, 5)$ . Find the equation of the curve.

Hint At a stationary point

\frac{d y}{d x} = 0

: that gives

k

first. Then integrate and use

(1, 5)

to find

c

B3
A curve has $\frac{d y}{d x} = k x - 3 x$ , where $k$ is a constant, and $x > 0$ . The curve has a stationary point at $(4, 10)$ . Find $k$ and the equation of the curve.
[5 marks]
Answer

Practice

A-Level

C1
Given that $\int (p x - \frac{3 q}{x ^{2}}) d x = 4 x^{\frac{3}{2}} + \frac{12}{x} + c$ , where $p$ and $q$ are constants, find $p$ and $q$ .
[3 marks]
Answer $p = \dots\dots$ $q = \dots\dots$

C2
Given that $y^{\frac{1}{2}} = 2 x^{\frac{1}{4}} - x^{- \frac{1}{4}}$ , $x > 0$ , find $\int y d x$ .
[4 marks]
Answer
C3
$f^{''} (x) = 12 x - 4$ . The curve $y = f (x)$ passes through $(1, 2)$ , and its gradient at that point is $3$ . Find $f (x)$ .
[4 marks]
Answer $f (x) = \dots\dots$

Exam-style questions

Edexcel exam style

D1
$f^{'} (x) = \frac{( x ^{2} - 3 ) ^{2}}{x ^{2}}, x \neq = 0$
(a)
Show that $f^{'} (x) = x^{2} - 6 + 9 x^{- 2}$ .
[2 marks]
(b)
Given that the point $(3, - 4)$ lies on the curve $y = f (x)$ , find $f (x)$ .
[4 marks]
Answer $f (x) = \dots\dots$
(c)
Find $f^{''} (x)$ . Hence show that the curve $y = f (x)$ has a stationary point where $x = 3$ , and that $f^{''} (3) = 0$ .
[4 marks]
Total for D1: 10 marks

D2
Leo writes: $\int \frac{6 x ^{3} - 5}{x ^{2}} d x = \frac{\frac{3}{2} x ^{4} - 5 x}{\frac{1}{3} x ^{3}} + c$
Leo is wrong.
(a)
Explain the mistake Leo has made, and find $\int \frac{6 x ^{3} - 5}{x ^{2}} d x$ correctly, giving each term in its simplest form.
[4 marks]
Answer
(b)
A curve has gradient $\frac{6 x ^{3} - 5}{x ^{2}}$ and passes through $(1, 2)$ . Hence find the equation of the curve.
[2 marks]
Answer $y = \dots\dots$
Total for D2: 6 marks
D3
The volume, $V$ cubic metres, of water in a tank $t$ hours after a pump is switched on is modelled by
$\frac{d V}{d t} = 24 t - 3 t$
When $t = 0$ , $V = 50$ . The model applies while the volume is increasing.
(a)
Find $V$ in terms of $t$ .
[4 marks]
Answer $V = \dots\dots$
(b)
Find the time at which the model stops applying. Hence find the greatest volume of water in the tank according to the model.
[4 marks]
Answer
Total for D3: 8 marks
D4
The curve $C$ has equation $y = f (x)$ , where $f^{'} (x) = 6 x^{2} + k x - 12$ and $k$ is a constant. $C$ passes through the points $(0, 5)$ and $(2, - 15)$ .
(a)
Find the value of $k$ and an expression for $f (x)$ .
[5 marks]
Answer
(b)
Find the coordinates of the stationary points of $C$ , and determine their nature.
[5 marks]
Answer
Total for D4: 10 marks

Extension

Stretch

No route is given. The hints are at the end of the sheet. Use one at a time.

E1
The polynomial $f (x)$ satisfies
$f^{'} (x) = 6 x^{2} - 2 f (1) x + f (2)$
for every $x$ , and $f (0) = 1$ . Find $f (x)$ .
[6 marks]
Hint 1 · What to try
$f (1)$ and $f (2)$ are just numbers. Call them $a$ and $b$ , and integrate.
Hint 2 · The first line
$f (x) = 2 x^{3} - a x^{2} + b x + 1$ . Now put $x = 1$ and $x = 2$ into this, remembering that $f (1) = a$ and $f (2) = b$ .
Hint 3 · The full method
$a = 2 - a + b + 1$ and $b = 16 - 4 a + 2 b + 1$ , so $2 a - b = 3$ and $4 a - b = 17$ . Then $a = 7$ and $b = 11$ .
Answer

How did it go?

Colour one circle on each line.

	Not yet	Nearly	Yes
I can integrate powers of x, including negative and fractional powers.
I can rewrite an expression in index form before integrating.
I can find a curve or a model from its rate of change and one point.

Answers and mark scheme

The worked answers and the full mark scheme are free with a Mathswariz account. Teachers and students can both sign up.

Get the answers