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Calculus

Calculus

Differentiate, then integrate ('hence')

Integration and definite integrals (14.10-14.12) Paper 1 + 2 7 min read

On this page
  1. In this note
  2. Formula bank
  3. When the integrand is the derivative (or a multiple)
  4. When the derivative has extra terms
  5. Ways they test it
  6. From the past papers
  7. Mistakes to avoid
  8. Summary
  9. Practice
  10. Can I do this?
  11. Quiz

In this note

Syllabus 14.10: integration as the reverse of differentiation
  • 14.10Use a derivative you have just found to integrate something you have no rule for.
  • 14.10Handle a multiple of the derivative, and rearrange when the derivative has extra terms.
  • 14.10Use the result with limits for an exact value.

Before you start: the product rule and chain rule, and e and trig integrals. Next: Finding a curve from its gradient.

Formula bank

The key factLearn itIf \(\dfrac{d}{dx}F(x)=g(x)\) then \(\displaystyle\int g(x)\,dx=F(x)+c\)Use it when 'hence', 'use your answer to part (a)'.
A multipleLearn it\(\displaystyle\int k\,g(x)\,dx=k\,F(x)+c\)Use it when the integrand is a number × the derivative.
Extra termsLearn itIf \(\frac{d}{dx}F=g+h\) and you can integrate h: \(\displaystyle\int g\,dx=F-\int h\,dx\)Use it when the derivative has an extra easy term.
Product ruleLearn it\(\dfrac{d}{dx}(uv)=u\dfrac{dv}{dx}+v\dfrac{du}{dx}\)you will usually need it in part (a).Use it when x ln x, x e^(kx), x sin kx.

Integrals like \(\int x\ln x\,dx\) or \(\int xe^{2x}\,dx\) have no rule in this syllabus. The exam always gives you a way in: first differentiate something, then 'hence' integrate. This page is the method for that.

When the integrand is the derivative (or a multiple)

General methodHow to answer a 'hence' question

  1. Do part (a): differentiate (product rule or chain rule). Simplify.
  2. Compare the integral you want with the derivative. Is it exactly the derivative, a multiple of it, or the derivative minus some easy terms?
  3. Integrate both sides of '\(\frac{d}{dx}F=\ldots\)': \(F=\int(\ldots)\,dx\).
  4. Rearrange so the integral you want is on its own; integrate any easy terms left; add + c.

Example Differentiate \((x^{2}+1)^{4}\). Hence find \(\int x(x^{2}+1)^{3}\,dx\).

Chain rule, then match the multiple.

  1. \(\frac{d}{dx}(x^{2}+1)^{4}=8x(x^{2}+1)^{3}\)
  2. so \(\int 8x(x^{2}+1)^{3}dx=(x^{2}+1)^{4}\)
  3. divide by 8

Answer \(\tfrac18(x^{2}+1)^{4}+c\)

Try it yourself

1
\(\frac{d}{dx}\sqrt{x^{2}+5}=\dfrac{x}{\sqrt{x^{2}+5}}\). Hence find \(\displaystyle\int\frac{3x}{\sqrt{x^{2}+5}}\,dx\).
Show answer

Answer \(3\sqrt{x^{2}+5}+c\)

2
\(\frac{d}{dx}(\ln x)^{2}=\dfrac{2\ln x}{x}\). Hence find \(\displaystyle\int\frac{\ln x}{x}\,dx\).
Show answer

Answer \(\tfrac12(\ln x)^{2}+c\)

When the derivative has extra terms

Product-rule derivatives come in two pieces. \(\frac{d}{dx}(xe^{3x})=e^{3x}+3xe^{3x}\). The piece you want (\(xe^{3x}\)) is stuck to a piece you CAN integrate (\(e^{3x}\)). Integrate the whole line, then move the easy piece across.

Example Given \(\frac{d}{dx}(xe^{3x})=e^{3x}+3xe^{3x}\), find \(\int xe^{3x}\,dx\).

Integrate both sides, then rearrange.

  1. \(xe^{3x}=\int e^{3x}dx+3\int xe^{3x}dx\)
  2. \(3\int xe^{3x}dx=xe^{3x}-\tfrac13e^{3x}\)

Answer \(\tfrac13xe^{3x}-\tfrac19e^{3x}+c\)

Example Differentiate \(x\ln x\). Hence find \(\int\ln x\,dx\).

\(\frac{d}{dx}(x\ln x)=\ln x+1\).

  1. \(x\ln x=\int\ln x\,dx+\int1\,dx\)

Answer \(\int\ln x\,dx=x\ln x-x+c\)

Try it yourself

1
\(\frac{d}{dx}(x\cos x)=\cos x-x\sin x\). Hence find \(\int x\sin x\,dx\).
Show answer

Answer \(\sin x-x\cos x+c\)

Ways they test it

Type 1 Exact multiple

"Find \(\frac{d}{dx}(x^{2}+5)^{3/2}\). Hence find \(\int x\sqrt{x^{2}+5}\,dx\)." (4 marks)

Spot it chain rule in (a).

  1. Spot the number you must divide by.
  2. + c.

Type 2 Product rule, rearrange

"Given \(y=x^{2}\ln x\), find \(\frac{dy}{dx}\). Hence find \(\int x\ln x\,dx\)." (5 marks)

Spot it product rule in (a).

  1. Integrate the whole derivative.
  2. Move the easy integral across.

For example \(2x\ln x+x\)

  1. \(x^{2}\ln x=2\int x\ln x+\int x\)

Answer \(\tfrac12x^{2}\ln x-\tfrac14x^{2}+c\)

Type 3 Then a definite integral

"Use your answer to (a) to find the exact value of \(\int_0^{\pi/6}2x\cos2x\,dx\)." (5 marks)

Spot it 'use your answer' + limits.

  1. Find the indefinite version first.
  2. Top − bottom, exact values.

From the past papers

Past paper 1

0606/12/M/J/24 Q11Paper 15 marksFull size
Given y = x^2 ln x find dy/dx, hence find the integral of x ln x.

How to start: (a) product rule with \(u=x^{2}\), \(v=\ln x\). (b) your derivative contains \(2x\ln x\): that's 2 × what you want.

✓ Show the solution have a go on paper first
  1. (a) \(\frac{dy}{dx}=2x\ln x+x^{2}\cdot\frac1x=2x\ln x+x\).
  2. (b) Integrate: \(x^{2}\ln x=2\int x\ln x\,dx+\int x\,dx\).
  3. (b) \(2\int x\ln x\,dx=x^{2}\ln x-\tfrac12x^{2}\).
  4. (b) answer: \(\tfrac12x^{2}\ln x-\tfrac14x^{2}+c\).
PartAnswer / workingMarksGuidance
(a)\(2x\ln x+x\)M1A1M1 for an attempt at the product rule
(b)\(2\int x\ln x\,dx=x^{2}\ln x-\int x\,dx\)M1dep on M1 in (a)
\(\tfrac12x^{2}\ln x\)A1
\(-\tfrac14x^{2}+c\)A1
Total: 5 marks

Examiner tip Keep the derivative from (a) in front of you and integrate EVERY term of it - leaving the 'x' behind loses the last two marks.

Past paper 2

0606/22/O/N/23 Q8Paper 2a, c7 marksFull size
A curve has equation y = x sin 2x. Find dy/dx, then use it to find the exact value of the integral of 2x cos 2x from 0 to pi/6.

How to start: (a) product rule. (c) your derivative is \(\sin2x+2x\cos2x\): integrate it, move the \(\sin2x\) part across.

✓ Show the solution have a go on paper first
  1. (a) \(\frac{dy}{dx}=\sin2x+2x\cos2x\).
  2. (c) \(x\sin2x=\int\sin2x\,dx+\int2x\cos2x\,dx\), so \(\int2x\cos2x\,dx=x\sin2x+\tfrac12\cos2x\).
  3. (c) \(\big[x\sin2x+\tfrac12\cos2x\big]_0^{\pi/6}=\tfrac{\pi}{6}\cdot\tfrac{\sqrt3}{2}+\tfrac12\cdot\tfrac12-\tfrac12\).
  4. (c) answer: \(\dfrac{\pi\sqrt3}{12}-\dfrac14\).
PartAnswer / workingMarksGuidance
(a)derivative of \(\sin2x\) is \(2\cos2x\)B1
\(2x\cos2x+\sin2x\)B1FT their 2cos2x
(c)\(x\sin2x+\tfrac12\cos2x\)M3M2 for \(x\sin2x+k\cos2x\); M1 for the rearranged integral
\(\tfrac{\pi}{6}\sin\tfrac{\pi}{3}+\tfrac12\cos\tfrac{\pi}{3}-\tfrac12\cos0\)A1
\(\tfrac{\pi\sqrt3}{12}-\tfrac14\)A1
Total: 7 marks

Examiner tip The lower limit is NOT zero here: \(\tfrac12\cos0=\tfrac12\) must be subtracted.

Mistakes to avoid

  • ✗ Integrating only the term you want✓ integrate the WHOLE derivative, then rearrange
  • ✗ Forgetting to divide by the multiple✓ \(\int8g=F\Rightarrow\int g=\tfrac18F\)
  • ✗ Lower limit taken as 0 when it isn't✓ evaluate \(F(0)\): \(\cos0=1\), \(e^{0}=1\)

Summary

  • \(\frac{d}{dx}F=g\Rightarrow\int g\,dx=F+c\).
  • Multiple: divide by the number.
  • Extra terms: integrate the whole derivative, move the easy integral across.

If you see …

  • 'hence' after a derivativereverse it.
  • x ln x, x e^(kx), x sin kxonly ever via 'hence'.

Practice

Practice: easy, medium, hard

Easy

1
\(\frac{d}{dx}(x^{2}+1)^{4}=8x(x^{2}+1)^{3}\). Find \(\int x(x^{2}+1)^{3}dx\).
Show answer

Answer \(\tfrac18(x^{2}+1)^{4}+c\)

2
\(\frac{d}{dx}(x\ln x)=\ln x+1\). Find \(\int\ln x\,dx\).
Show answer

Answer \(x\ln x-x+c\)

3
\(\frac{d}{dx}(xe^{3x})=e^{3x}+3xe^{3x}\). Find \(\int(e^{3x}+3xe^{3x})dx\).
Show answer

Answer \(xe^{3x}+c\)

Medium

1
\(\frac{d}{dx}(xe^{-x})=e^{-x}-xe^{-x}\). Find \(\int xe^{-x}dx\).
Guide me

What is it asking? an integral

What do we know? \(xe^{-x}=\int e^{-x}-\int xe^{-x}\)

Plan rearrange

Show working
  1. \(-xe^{-x}-e^{-x}+c\)

Answer \(-xe^{-x}-e^{-x}+c\)

2
\(\frac{d}{dx}(x\cos x)=\cos x-x\sin x\). Find \(\int x\sin x\,dx\).
Guide me

What is it asking? an integral

What do we know? \(x\cos x=\sin x-\int x\sin x\)

Plan rearrange

Show working
  1. \(\sin x-x\cos x+c\)

Answer \(\sin x-x\cos x+c\)

3
\(\frac{d}{dx}\big(x\sin\tfrac x2\big)=\sin\tfrac x2+\tfrac x2\cos\tfrac x2\). Find \(\int x\cos\tfrac x2\,dx\).
Guide me

What is it asking? an integral

What do we know? \(x\sin\tfrac x2=-2\cos\tfrac x2+\tfrac12\int x\cos\tfrac x2\)

Plan × 2

Show working
  1. \(2x\sin\tfrac x2+4\cos\tfrac x2+c\)

Answer \(2x\sin\tfrac x2+4\cos\tfrac x2+c\)

Hard

1
Find the exact value of \(\int_0^1 xe^{3x}\,dx\). (Type (2e^3+1)/9.)
Guide me

What is it asking? a value

What do we know? \(\tfrac13xe^{3x}-\tfrac19e^{3x}\)

Plan top − bottom

Show working
  1. \(\tfrac13e^{3}-\tfrac19e^{3}+\tfrac19\)

Answer \(\tfrac{2e^{3}+1}{9}\)

PartAnswer / workingMarksGuidance
indefinite integralM1A1
limitsM1
answerA1
Total: 4 marks
2
Given \(y=x^{2}\ln x\), find \(\int x\ln x\,dx\).
Guide me

What is it asking? an integral

What do we know? \(\frac{dy}{dx}=2x\ln x+x\)

Plan integrate, rearrange, ÷ 2

Show working
  1. \(\tfrac12x^{2}\ln x-\tfrac14x^{2}+c\)

Answer \(\tfrac12x^{2}\ln x-\tfrac14x^{2}+c\)

PartAnswer / workingMarksGuidance
rearrangedM1
each termA1A1
Total: 3 marks
3
Exact value of \(\int_0^{\pi/6}2x\cos2x\,dx\). (Type pi*sqrt(3)/12-1/4.)
Guide me

What is it asking? a value

What do we know? \(x\sin2x+\tfrac12\cos2x\)

Plan at \(\tfrac{\pi}{6}\) minus at 0

Show working
  1. \(\tfrac{\pi\sqrt3}{12}-\tfrac14\)

Answer \(\tfrac{\pi\sqrt3}{12}-\tfrac14\)

PartAnswer / workingMarksGuidance
integralM3
limitsA1
answerA1
Total: 5 marks

Can I do this?

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

Quiz

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