Back to course
11.1PureCore

Integrating Standard Functions

Integration reverses differentiation, so every derivative you have learned gives a standard integral. This lesson collects the key results for trig, exponential and reciprocal functions.

24 min Integration
Edexcel A level Maths: 11.1 Integrating Standard FunctionsCurated by Kepler, not produced by us. Watch it before the notes below.
Open on YouTube

What you'll be able to do

  • Integrate eˣ, 1/x and trig functions
  • Recognise integrals as reverse derivatives
  • Include the constant of integration
  • Apply to definite integrals
1

The standard integrals

Each follows from a known derivative (remember the +c+c):

∫ex dx=ex+c\int e^x\,dx = e^x + c
Exponential.
∫1x dx=ln⁡∣x∣+c\int \frac{1}{x}\,dx = \ln|x| + c
Reciprocal (note the modulus).
∫cos⁡x dx=sin⁡x+c,∫sin⁡x dx=−cos⁡x+c\int \cos x\,dx = \sin x + c, \quad \int \sin x\,dx = -\cos x + c
Trig.
∫sec⁡2x dx=tan⁡x+c\int \sec^2 x\,dx = \tan x + c
From d/dx(tan x).
2

Worked example

These are read straight off the differentiation results.

Example 1Find ∫(ex+cos⁡x) dx\int (e^x + \cos x)\,dx.
1∫ex dx=ex\int e^x\,dx = e^x, ∫cos⁡x dx=sin⁡x\int \cos x\,dx = \sin x.
2Add +c+c.
Answer
ex+sin⁡x+ce^x + \sin x + c

Tip — For 1/x the answer is ln|x| + c — keep the modulus.

Formula recap

∫ex dx=ex+c\int e^x\,dx = e^x + c
Exponential.
∫1x dx=ln⁡∣x∣+c\int \tfrac{1}{x}\,dx = \ln|x| + c
Reciprocal.
∫sin⁡x dx=−cos⁡x+c\int \sin x\,dx = -\cos x + c
Sine (note the minus).

Common mistakes to avoid

Writing ∫1/x dx = ln x + c.
It is ln|x| + c.
Forgetting +c on indefinite integrals.
Always include the constant of integration.

Key takeaways

  • ∫eˣ = eˣ + c; ∫1/x = ln|x| + c.
  • ∫cos x = sin x + c; ∫sin x = −cos x + c.
  • Every standard integral is a reverse derivative.

Test yourself

Answer a few questions, then review anything you missed.

Prefer paper? Open the written self-check

Answer each question before opening its answer. This self-check does not record a quiz score.

  1. ∫ex dx\int e^x\,dx equals:

    • ex+ce^x + c
    • xex+cx e^x + c
    • exx+c\frac{e^x}{x} + c
    • exe^x
    Show answer and explanation

    Answer: ex+ce^x + c

    Reverse derivative.

  2. ∫1x dx\int \frac{1}{x}\,dx equals:

    • ln⁡∣x∣+c\ln|x| + c
    • ln⁡x\ln x
    • −1x2+c-\frac{1}{x^2} + c
    • x−1+cx^{-1}+c
    Show answer and explanation

    Answer: ln⁡∣x∣+c\ln|x| + c

    Modulus needed.

  3. ∫sin⁡x dx\int \sin x\,dx equals:

    • −cos⁡x+c-\cos x + c
    • cos⁡x+c\cos x + c
    • −sin⁡x+c-\sin x + c
    • sin⁡x\sin x
    Show answer and explanation

    Answer: −cos⁡x+c-\cos x + c

    Reverse of d/dx(cos)=−sin.

  4. ∫sec⁡2x dx\int \sec^2 x\,dx equals:

    • tan⁡x+c\tan x + c
    • sec⁡x+c\sec x + c
    • cot⁡x+c\cot x + c
    • sec⁡2x\sec^2 x
    Show answer and explanation

    Answer: tan⁡x+c\tan x + c

    From d/dx(tan).

  5. ∫cos⁡x dx\int \cos x\,dx equals:

    • sin⁡x+c\sin x + c
    • −sin⁡x+c-\sin x + c
    • cos⁡x+c\cos x + c
    • −cos⁡x+c-\cos x + c
    Show answer and explanation

    Answer: sin⁡x+c\sin x + c

    Reverse derivative.

Next step

Keep practising

Start with the quiz above to check the key ideas. You can also choose another topic from the practice library.

Browse practice topics

Related lessons

All 11 lessons

Put the methods into practice: see how A-Level maths duels work. An account is required to play; the overview and sample are public.