Calculus

Definite integrals and unknown limits

Integration and definite integrals (14.10-14.12) Paper 1 + 2 6 min read

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In this note
Formula bank
Evaluating a definite integral
Finding an unknown limit
Ways they test it
From the past papers
Mistakes to avoid
Summary
Practice
Can I do this?
Quiz

In this note

Syllabus 14.13: definite integrals

14.13Evaluate a definite integral \(\int_a^b f(x)\,dx\), giving an exact answer when asked.
14.13Combine logs into a single log using the log laws.
14.13Find an unknown limit when the value of the integral is given.

Before you start: Integration basics, e and trig, log laws and solving quadratics. Next: Differentiate, then integrate.

Formula bank

Definite integralLearn it\(\displaystyle\int_a^b f(x)\,dx=\big[F(x)\big]_a^b=F(b)-F(a)\)F = any integral of f; no + c (it cancels).Use it when limits on the integral sign.

Swap the limitsLearn it\(\displaystyle\int_b^a f\,dx=-\int_a^b f\,dx\)Use it when limits the 'wrong' way round.

Log lawsLearn it\(\ln p+\ln q=\ln pq\); \(\ln p-\ln q=\ln\frac pq\); \(k\ln p=\ln p^{k}\)Use it when 'as a single logarithm', 'in the form ln a'.

Exact valuesLearn it\(\ln1=0\), \(e^{0}=1\), \(e^{\ln k}=k\), \(\sin\tfrac{\pi}{6}=\tfrac12\), \(\tan\tfrac{\pi}{4}=1\)Use it when Paper 1 (no calculator).

Evaluating a definite integral

General methodHow to evaluate any definite integral

Integrate as usual, but leave out + c. Write it in square brackets with the limits.
Substitute the TOP limit, then the BOTTOM limit.
Top − bottom. Keep logs, surds, e and π exact unless a decimal is asked for.

Example Evaluate \(\displaystyle\int_1^4\left(3\sqrt x-\frac{2}{x^{2}}\right)dx\).

Integrate, then top − bottom.

\(\big[2x^{3/2}+2x^{-1}\big]_1^4\)
\((16+0.5)-(2+2)\)

Answer 12.5

Example Find the exact value of \(\displaystyle\int_1^3\left(\frac1x+\frac{2}{2x-1}\right)dx\).

Two logs, then one log.

\(\big[\ln x+\ln(2x-1)\big]_1^3\)
\((\ln3+\ln5)-(\ln1+\ln1)\)

Answer \(\ln15\)

Try it yourself

Evaluate \(\int_0^1(x^{3}+2x)\,dx\).

Show answer

Answer 1.25

Find the exact value of \(\displaystyle\int_2^5\frac{1}{x-1}\,dx\). (Type ln(4).)

Show answer

Answer \(\ln4\)

Finding an unknown limit

Sometimes they give you the VALUE of the integral and a letter for a limit. Integrate and substitute exactly as before, set the result equal to the value, and solve. A log equation becomes an ordinary equation once both sides are a single log; an exponential equation is solved with ln.

General methodHow to find an unknown limit

Integrate; substitute the letter and the number; top − bottom.
Set this equal to the given value.
Logs: make each side one log, then drop the ln. Exponentials: isolate \(e^{\ldots}\), then take ln.
Solve; reject values outside the condition given (such as a > 1).

Example Given \(\int_0^a e^{x/2}\,dx=4\), find the exact value of a.

\(\int e^{x/2}dx=2e^{x/2}\).

\(2e^{a/2}-2=4\)
\(e^{a/2}=3\)
\(\tfrac a2=\ln3\)

Answer \(a=2\ln3\)

Example Given \(\int_1^k(2x-3)\,dx=4\) where \(k>1\), find k.

A quadratic in k.

\(\big[x^{2}-3x\big]_1^k=k^{2}-3k+2\)
\(k^{2}-3k-2=0\)
\(k=\dfrac{3\pm\sqrt{17}}{2}\); \(k>1\)

Answer \(k=\dfrac{3+\sqrt{17}}{2}\)

Try it yourself

Given \(\int_0^{a}e^{2x}\,dx=4\), find a exactly. (Type ln(3) etc.)

Hint

\(\tfrac12(e^{2a}-1)=4\).

Show answer

Answer \(\ln3\)

Ways they test it

Type 1 Evaluate

"Find the exact value of \(\int_1^3\ldots dx\)." (3-5 marks)

Spot it limits, 'exact'.

Integrate; substitute; top − bottom.
Simplify logs/surds.

Type 2 Single log / given form

"…giving your answer in the form \(\ln a+b\)." (5 marks)

Spot it a stated form.

Collect the ln terms with the log laws.
Numbers that aren't logs make b.

Type 3 Unknown limit, logs

"\(\int_1^a(\ldots)dx=\ln\tfrac15\). Find a." (6 marks)

Spot it a letter limit, ln on the right.

Single log both sides.
Quadratic in a; reject the root that breaks the condition.

For example answer gives \(a^{2}-4a-3=0\)

\(a=2\pm\sqrt7\)

Answer \(a=2+\sqrt7\)

Type 4 Unknown limit, e

"\(\int_0^a e^{2x}dx=50\). Find a." (4 marks)

Spot it e and a value.

Isolate \(e^{2a}\).
Take ln.

From the past papers

Past paper 1

Given that the integral from 1 to a of 3/(3x+2) - 2/(2x+1) - 1/x is ln(1/5), find a. — 4037/13/O/N/22 Q11Paper 16 marksFull size

How to start: Each term is a log. Collect into ONE log, set it equal to \(\ln\tfrac15\), then drop the ln.

✓ Show the solution have a go on paper first

\(\big[\ln(3x+2)-\ln(2x+1)-\ln x\big]_1^a\).
\(=\ln\dfrac{3a+2}{a(2a+1)}-\ln\dfrac53=\ln\dfrac{3(3a+2)}{5a(2a+1)}\).
\(\dfrac{3(3a+2)}{5a(2a+1)}=\dfrac15\Rightarrow 9a+6=2a^{2}+a\Rightarrow a^{2}-4a-3=0\).
\(a=2\pm\sqrt7\); since \(a>1\), \(a=2+\sqrt7\).

Part	Answer / working	Marks	Guidance
	\(\ln(3x+2)-\ln(2x+1)-\ln x\)	B2	B1 for one correct term, B1 for the other two
	correct use of limits	M1	dep on a B mark
	log laws to a single log	M1
	\(a^{2}-4a-3=0\) and solve	M1
	\(a=2+\sqrt7\) only	A1	must reject \(2-\sqrt7\)
Total: 6 marks

Examiner tip Write \(\ln\tfrac15=-\ln5\) only if it helps; the neat route is one log on each side. Always reject the root that breaks a > 1.

Past paper 2

Given that the integral from 0 to a of e^(2x) is 50, find the exact value of a. — 4037/21/M/J/19 Q12Paper 2a4 marksFull size

How to start: \(\int e^{2x}dx=\tfrac12e^{2x}\); substitute a and 0; then ln.

✓ Show the solution have a go on paper first

\(\big[\tfrac12e^{2x}\big]_0^a=\tfrac12e^{2a}-\tfrac12=50\).
\(e^{2a}=101\).
\(2a=\ln101\Rightarrow a=\tfrac12\ln101\).

Part	Answer / working	Marks	Guidance
(a)	\(\tfrac12e^{2x}\)	B1
	\(\tfrac12e^{2a}-\tfrac12=50\)	M1	limits used, set = 50
	\(2a=\ln101\)	M1	rearrange and take ln
	\(a=\tfrac12\ln101\)	A1	any exact equivalent
Total: 4 marks

Examiner tip Forgetting \(e^{0}=1\) (writing the lower limit as 0) is the usual slip.

Mistakes to avoid

✗ Bottom − top✓ TOP − bottom
✗ Treating \(e^{0}\) as 0✓ \(e^{0}=1\)
✗ \(\ln1\) left in, or \(\ln(a+b)=\ln a+\ln b\)✓ \(\ln1=0\); only \(\ln(ab)=\ln a+\ln b\)
✗ A decimal when 'exact' was asked✓ keep ln, √, e, πThe answer mark needs the exact form.

Summary

\(\int_a^b f\,dx=F(b)-F(a)\), no + c.
Exact: keep logs, surds, e and π; combine logs with the log laws.
Unknown limit: integrate, substitute, set equal, solve, reject impossible roots.

If you see …

'exact value'no decimals.
a letter as a limitmake an equation.

Practice

Practice: easy, medium, hard

Easy

\(\int_0^2 3x^{2}\,dx\)

Show answer

Answer 8

\(\int_0^{\pi/4}\sec^{2}x\,dx\)

Show answer

Answer 1

\(\int_0^4\frac{1}{\sqrt{2x+1}}\,dx\)

Show answer

Answer 2

Medium

Evaluate \(\int_1^4\big(3\sqrt x-\frac{2}{x^{2}}\big)dx\).

Guide me

What is it asking? a value

What do we know? \(2x^{3/2}+2x^{-1}\)

Plan 16.5 − 4

Show working

12.5

Answer 12.5

Exact value of \(\int_0^2\frac{6}{3x+1}\,dx\). (Type 2ln(7).)

Guide me

What is it asking? a value

What do we know? \(2\ln(3x+1)\)

Plan \(2\ln7-2\ln1\)

Show working

\(2\ln7\)

Answer \(2\ln7\)

Exact value of \(\int_0^{\pi/6}\cos2x\,dx\). (Type sqrt(3)/4.)

Guide me

What is it asking? a value

What do we know? \(\tfrac12\sin2x\)

Plan \(\tfrac12\sin\tfrac{\pi}{3}\)

Show working

\(\tfrac{\sqrt3}4\)

Answer \(\tfrac{\sqrt3}{4}\)

Hard

\(\int_1^k(2x-3)\,dx=4\), \(k>1\). Find k exactly. (Type (3+sqrt(17))/2.)

Guide me

What is it asking? a value

What do we know? \(k^{2}-3k+2=4\)

Plan quadratic formula, k > 1

Show working

\(\tfrac{3+\sqrt{17}}{2}\)

Answer \(\tfrac{3+\sqrt{17}}2\)

Part	Answer / working	Marks
	\(x^{2}-3x\)	B1
	\(k^{2}-3k-2=0\)	M1
	answer	A1
Total: 3 marks

\(\int_0^a e^{x/2}\,dx=4\). Find a exactly.

Guide me

What is it asking? a value

What do we know? \(2e^{a/2}-2=4\)

Plan \(e^{a/2}=3\)

Show working

\(2\ln3\)

Answer \(2\ln3\)

Part	Answer / working	Marks
	\(2e^{x/2}\)	B1
	limits, = 4	M1
	\(2\ln3\)	A1
Total: 3 marks

\(\int_1^b(2x-2)\,dx=3\), \(b>1\). Find b exactly.

Guide me

What is it asking? a value

What do we know? \(b^{2}-2b+1=3\)

Plan \((b-1)^{2}=3\)

Show working

\(1+\sqrt3\)

Answer \(1+\sqrt3\)

Part	Answer / working	Marks
	equation in b	M1
	\(1+\sqrt3\)	A1
Total: 2 marks

Can I do this?

I can evaluate a definite integral.
I can give exact answers and combine logs.
I can find an unknown limit.

Anything unticked: revise that section first. All ticked: take the quiz.

Quiz

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