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.

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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

The standard integrals

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

\int e^{x} d x = e^{x} + c

Exponential.

\int \frac{1}{x} d x = ln ∣ x ∣ + c

Reciprocal (note the modulus).

\int cos x d x = sin x + c, \int sin x d x = - cos x + c

Trig.

\int sec^{2} x d x = tan x + c

From d/dx(tan x).

Worked example

These are read straight off the differentiation results.

\int e^{x} d x = e^{x}

\int cos x d x = sin x

2Add

+ c

Answer

e^{x} + sin x + c

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

Formula recap

\int e^{x} d x = e^{x} + c

Exponential.

\int \frac{1}{x} d x = ln ∣ x ∣ + c

Reciprocal.

\int sin x d x = - 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.

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Integrating Standard Functions

The standard integrals

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Integrating Standard Functions

The standard integrals

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