1.08b–cPureCore

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.

30 min Integration

The big picture

For a linear inner expression, divide by its derivative when reversing differentiation. Thus $\int e^{ax+b}dx=e^{ax+b}/a+C$ for nonzero a. For $\int (ax+b)^{-1}dx$ , use $(1/a)\ln|ax+b|+C$ , rather than the power rule. Check an antiderivative by differentiating it. This immediately detects a missed inner coefficient or a sign error in a sine/cosine integral. State a domain interval when a logarithm or denominator imposes a restriction.

Edexcel A level Maths: 11.1 Integrating Standard FunctionsCurated by Kepler, not produced by us. Watch it, then work through 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\,dx = e^x + c

Exponential.

\int \frac{1}{x}\,dx = \ln|x| + c

Reciprocal (note the modulus).

\int \cos x\,dx = \sin x + c, \quad \int \sin x\,dx = -\cos x + c

Trig.

\int \sec^2 x\,dx = \tan x + c

From d/dx(tan x).

Worked example

These are read straight off the differentiation results.

Example 1Find

\int (e^x + \cos x)\,dx

\int e^x\,dx = e^x

\int \cos x\,dx = \sin x

2Add

+c

Answer

e^x + \sin x + c

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

Linear inner functions and the logarithm exception

Check an antiderivative by differentiating it. This immediately detects a missed inner coefficient or a sign error in a sine/cosine integral. State a domain interval when a logarithm or denominator imposes a restriction.

Example 1Find

\int\left(6e^{2x}-4\sin3x+\dfrac5{2x+1}\right)dx

1The exponential gives

3e^{2x}

2Since

d(\cos3x)/dx=-3\sin3x

, the trig term gives

(4/3)\cos3x

3The reciprocal linear term gives

(5/2)\ln|2x+1|

4Add one arbitrary constant and differentiate to verify each term.

Answer

3e^{2x}+\frac43\cos3x+\frac52\ln|2x+1|+C

, on intervals excluding

x=-1/2

Think like an examiner

Write the defining equations and show the substitution before evaluating.
Check the domain, units and interpretation of the final result.

Common misconceptions

Writing ∫1/x dx = ln x + c.

It is ln|x| + c.

Forgetting +c on indefinite integrals.

Always include the constant of integration.

Key results to remember

\int e^x\,dx = e^x + c

Exponential.

\int \tfrac{1}{x}\,dx = \ln|x| + c

Reciprocal.

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

Sine (note the minus).

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.

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

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