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Calculus

Calculus

Integrating exponentials and trig

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

On this page
  1. In this note
  2. Formula bank
  3. Exponentials
  4. Sin, cos and sec²
  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 14.12: exponentials and trig
  • 14.12Integrate \(e^{ax+b}\).
  • 14.12Integrate \(\sin(ax+b)\), \(\cos(ax+b)\) and \(\sec^{2}(ax+b)\) (angles in radians).
  • 14.12Use an identity to turn \(\tan^{2}x\), \((\sin x+\cos x)^{2}\) or \(\sin x\cos x\) into these forms.

Before you start: Integration basics; derivatives of \(e^{x}\), \(\sin x\), \(\cos x\), \(\tan x\). Next: Definite integrals.

Formula bank

ExponentialLearn it\(\displaystyle\int e^{ax+b}\,dx=\frac1a e^{ax+b}+c\)e stays e; divide by a.Use it when \(e^{2x}\), \(e^{1-2x}\).
CosineLearn it\(\displaystyle\int\cos(ax+b)\,dx=\frac1a\sin(ax+b)+c\)Use it when cos.
SineLearn it\(\displaystyle\int\sin(ax+b)\,dx=-\frac1a\cos(ax+b)+c\)note the minus.Use it when sin.
Sec squaredLearn it\(\displaystyle\int\sec^{2}(ax+b)\,dx=\frac1a\tan(ax+b)+c\)Use it when \(\sec^{2}\).
IdentitiesLearn it\(\tan^{2}x=\sec^{2}x-1\); \(\sin x\cos x=\tfrac12\sin2x\); \((\sin x+\cos x)^{2}=1+\sin2x\)Use it when \(\tan^{2}\), products of sin and cos.

Exponentials

\(e^{x}\) is its own derivative, so it is its own integral. With \(e^{ax+b}\) the chain rule multiplies by a when you differentiate, so you DIVIDE by a when you integrate.

General methodHow to integrate any exponential

  1. Expand or split until every term is \(k\,e^{ax+b}\) (e.g. \((e^{x}+2)^{2}=e^{2x}+4e^{x}+4\); \(\tfrac{1}{e^{x}}=e^{-x}\)).
  2. Each term: keep \(e^{ax+b}\), divide by a.
  3. Constants integrate to (number) × x. Add + c.

Example Find \(\int(4e^{2x}-3e^{-x})\,dx\).

Divide by 2, then by −1.

  1. \(\tfrac42e^{2x}-\tfrac{3}{-1}e^{-x}\)

Answer \(2e^{2x}+3e^{-x}+c\)

Example Find \(\int(e^{x}+2)^{2}\,dx\).

Expand first.

  1. \(e^{2x}+4e^{x}+4\)

Answer \(\tfrac12e^{2x}+4e^{x}+4x+c\)

Try it yourself

1
Find \(\int e^{3-4x}\,dx\). (Type e^(3-4x).)
Show answer

Answer \(-\tfrac14e^{3-4x}+c\)

2
Find \(\int 3e^{6x}\,dx\).
Show answer

Answer \(\tfrac12e^{6x}+c\)

Sin, cos and sec²

Read the derivative table backwards: \(\sin\to\cos\), \(\cos\to-\sin\), \(\tan\to\sec^{2}\). So \(\int\cos=\sin\), \(\int\sin=-\cos\), \(\int\sec^{2}=\tan\), and divide by a for \((ax+b)\). Angles are ALWAYS in radians in calculus.

General methodHow to integrate any trig term

  1. Is it \(\sin(ax+b)\), \(\cos(ax+b)\) or \(\sec^{2}(ax+b)\)? If not, use an identity first.
  2. \(\cos\to\sin\); \(\sin\to-\cos\); \(\sec^{2}\to\tan\).
  3. Divide by a. Add + c.

Example Find \(\int\big(6\cos3x-2\sin\tfrac x2\big)\,dx\).

a = 3 and a = ½.

  1. \(\tfrac63\sin3x\)
  2. \(-2\times\big(-\tfrac{1}{1/2}\big)\cos\tfrac x2=4\cos\tfrac x2\)

Answer \(2\sin3x+4\cos\tfrac x2+c\)

Example Find \(\int\tan^{2}x\,dx\).

\(\tan^{2}x\) is not on the list: use \(\tan^{2}x=\sec^{2}x-1\).

  1. \(\int(\sec^{2}x-1)\,dx\)

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

Try it yourself

1
Find \(\int 8\sin4x\,dx\).
Show answer

Answer \(-2\cos4x+c\)

2
Find \(\int(\sin x+\cos x)^{2}\,dx\).
Hint
\(\sin^2x+\cos^2x+2\sin x\cos x=1+\sin2x\).
Show answer

Answer \(x-\tfrac12\cos2x+c\)

Ways they test it

Type 1 Mixed indefinite

"Find \(\int\big(\tfrac{1}{4x-3}+\tfrac1{x^{3}}\big)dx\)." (3 marks)

Spot it two or three different kinds of term.

  1. Each term with its own rule.
  2. + c once.

Type 2 Trig with limits

"Evaluate \(\int_{\pi/3}^{\pi/2}\cos\tfrac x4\,dx\)." (4 marks)

Spot it radian limits.

  1. Integrate: \(4\sin\tfrac x4\).
  2. Top − bottom (calculator in RADIANS, or exact values).

Type 3 Identity first

"Find \(\int\tan^{2}3x\,dx\)." (3 marks)

Spot it \(\tan^{2}\), \(\sin x\cos x\), a squared bracket of trig.

  1. \(\tan^{2}=\sec^{2}-1\) etc.
  2. Then integrate.

For example \(\tan^{2}x\)

  1. \(\sec^{2}x-1\)

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

Type 4 e with a letter limit

"Find, in terms of a, \(\int_{0.5}^{a}e^{1-2x}dx\)." (3 marks)

Spot it a letter as a limit.

  1. Integrate.
  2. Substitute a and the number; simplify.

From the past papers

Past paper 1

4037/22/M/J/24 Q2Paper 27 marksFull size
Evaluate the integral of cos(x/4) from pi/3 to pi/2, and find the integral of 1/(4x-3) + 1/x^3.

How to start: (a) cos with \(a=\tfrac14\): dividing by \(\tfrac14\) means × 4. Calculator in radians. (b) two separate rules: a log and a power.

✓ Show the solution have a go on paper first
  1. (a) \(\int\cos\tfrac x4\,dx=4\sin\tfrac x4\).
  2. (a) \(4\sin\tfrac{\pi}{8}-4\sin\tfrac{\pi}{12}=1.5307-1.0353\).
  3. (a) answer: \(0.495\) (3 s.f.).
  4. (b) \(\int\frac{1}{4x-3}dx=\tfrac14\ln(4x-3)\); \(\int x^{-3}dx=\frac{x^{-2}}{-2}\).
  5. (b) answer: \(\tfrac14\ln(4x-3)-\dfrac{1}{2x^{2}}+c\).
PartAnswer / workingMarksGuidance
(a)\(4\sin\tfrac x4\)B2B1 for \(k\sin\tfrac x4\)
\(4\sin\tfrac{\pi}{8}-4\sin\tfrac{\pi}{12}\)M1correct use of limits
0.495A1
(b)\(\tfrac14\ln(4x-3)\)B2B1 for \(k\ln(4x-3)\)
\(-\tfrac{1}{2x^{2}}\) (+ c)B1
Total: 7 marks

Examiner tip A calculator left in degrees gives a wrong (a). Write the integral before substituting, or the method mark is hard to give.

Past paper 2

4037/23/O/N/24 Q6Paper 2b3 marksFull size
Find, in terms of a, the integral of e^(1-2x) from 0.5 to a.

How to start: Exponential with \(a=-2\): divide by −2.

✓ Show the solution have a go on paper first
  1. \(\int e^{1-2x}dx=-\tfrac12e^{1-2x}\).
  2. \(\big[-\tfrac12e^{1-2x}\big]_{0.5}^{a}=-\tfrac12e^{1-2a}+\tfrac12e^{0}\).
  3. Answer: \(\tfrac12-\tfrac12e^{1-2a}\).
PartAnswer / workingMarksGuidance
(b)\(-\tfrac12e^{1-2x}\)B2B1 for \(ke^{1-2x}\)
\(\tfrac12-\tfrac12e^{1-2a}\) iswB1
Total: 3 marks

Mistakes to avoid

  • ✗ \(\int\sin x\,dx=\cos x\)✓ \(-\cos x\)
  • ✗ \(\int e^{2x}dx=2e^{2x}\)✓ \(\tfrac12e^{2x}\)Divide, don't multiply.
  • ✗ \(\int e^{x}dx=\frac{e^{x+1}}{x+1}\)✓ \(e^{x}\)The power rule is only for x to a number.
  • ✗ \(\int\tan^{2}x\,dx=\frac{\tan^{3}x}{3}\)✓ use \(\sec^{2}x-1\)

Summary

  • \(e^{ax+b}\to\frac1ae^{ax+b}\).
  • \(\cos\to\sin\), \(\sin\to-\cos\), \(\sec^{2}\to\tan\); divide by a; radians.
  • \(\tan^{2}=\sec^{2}-1\), \(\sin x\cos x=\tfrac12\sin2x\).

If you see …

  • e or trig of (ax + b)divide by a.
  • \(\tan^{2}\) or a productidentity first.

Practice

Practice: easy, medium, hard

Easy

1
\(\int\cos\tfrac x3\,dx\)
Show answer

Answer \(3\sin\tfrac x3+c\)

2
\(\int\sec^{2}5x\,dx\)
Show answer

Answer \(\tfrac15\tan5x+c\)

3
\(\int 2\sin x\cos x\,dx\)
Show answer

Answer \(-\tfrac12\cos2x+c\)

Medium

1
\(\int\big(3\sin(2x-1)+2e^{2x-1}\big)dx\)
Guide me

What is it asking? an integral

What do we know? divide by 2 each

Plan sin → −cos

Show working
  1. \(-\tfrac32\cos(2x-1)+e^{2x-1}+c\)

Answer \(-\tfrac32\cos(2x-1)+e^{2x-1}+c\)

2
\(\int(e^{2x}+e^{-2x})^{2}\,dx\)
Guide me

What is it asking? an integral

What do we know? \(e^{4x}+2+e^{-4x}\)

Plan each term

Show working
  1. \(\tfrac14e^{4x}+2x-\tfrac14e^{-4x}+c\)

Answer \(\tfrac14e^{4x}+2x-\tfrac14e^{-4x}+c\)

3
\(\int 4\tan^{2}2x\,dx\)
Guide me

What is it asking? an integral

What do we know? \(4\sec^{2}2x-4\)

Plan integrate

Show working
  1. \(2\tan2x-4x+c\)

Answer \(2\tan2x-4x+c\)

Hard

1
Evaluate \(\int_{\pi/3}^{\pi/2}\cos\tfrac x4\,dx\) (3 s.f.).
Guide me

What is it asking? a value

What do we know? \(4\sin\tfrac x4\)

Plan top − bottom in radians

Show working
  1. 0.495

Answer 0.495

PartAnswer / workingMarksGuidance
\(4\sin\tfrac x4\)B2
limitsM1
0.495A1
Total: 4 marks
2
Find \(\int_{0.5}^{a}e^{1-2x}\,dx\) in terms of a. (Type e^(1-2a).)
Guide me

What is it asking? an expression

What do we know? \(-\tfrac12e^{1-2x}\)

Plan a then 0.5

Show working
  1. \(\tfrac12-\tfrac12e^{1-2a}\)

Answer \(\tfrac12-\tfrac12e^{1-2a}\)

PartAnswer / workingMarksGuidance
\(-\tfrac12e^{1-2x}\)B2
answerB1
Total: 3 marks
3
\(\int(\sin x+\cos x)^{2}\,dx\)
Guide me

What is it asking? an integral

What do we know? \(1+\sin2x\)

Plan integrate

Show working
  1. \(x-\tfrac12\cos2x+c\)

Answer \(x-\tfrac12\cos2x+c\)

PartAnswer / workingMarksGuidance
\(1+\sin2x\)M1
answerA1
Total: 2 marks

Can I do this?

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

Quiz

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