Integration can be used to find areas, volumes, central points and many useful things. It is often used to find the area underneath the graph of a function and the x-axis.

The first rule to know is that integrals and derivatives are opposites!

Sometimes we can work out an integral,because we know a matching derivative.

### Integration Rules

Here are the most useful rules, with examples below:

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Common FunctionsFunctionIntegralRulesFunctionIntegral
Constant∫a dxax + C
Variable∫x dxx2/2 + C
Square∫x2 dxx3/3 + C
Reciprocal∫(1/x) dxln|x| + C
Exponential∫ex dxex + C
∫ax dxax/ln(a) + C
∫ln(x) dxx ln(x) − x + C
Trigonometry (x in radians)∫cos(x) dxsin(x) + C
∫sin(x) dx-cos(x) + C
∫sec2(x) dxtan(x) + C
Multiplication by constant∫cf(x) dxc∫f(x) dx
Power Rule (n≠−1)∫xn dxxn+1n+1 + C
Sum Rule∫(f + g) dx∫f dx + ∫g dx
Difference Rule∫(f - g) dx∫f dx - ∫g dx
Integration by PartsSee Integration by Parts
Substitution RuleSee Integration by Substitution

### Example: what is the integral of sin(x) ?

From the table above it is listed as being −cos(x) + C

It is written as:

∫sin(x) dx = −cos(x) + C

### Example: what is the integral of 1/x ?

From the table above it is listed as being ln|x| + C

It is written as:

∫(1/x) dx = ln|x| + C

The vertical bars || either side of x mean absolute value, because we don"t want to give negative values to the natural logarithm function ln.

### Example: What is ∫x3 dx ?

The question is asking "what is the integral of x3 ?"

We can use the Power Rule, where n=3:

∫xn dx = xn+1n+1 + C

∫x3 dx = x44 + C

### Example: What is ∫√x dx ?

√x is also x0.5

We can use the Power Rule, where n=0.5:

∫xn dx = xn+1n+1 + C

∫x0.5 dx = x1.51.5 + C

### Example: What is ∫6x2 dx ?

We can move the 6 outside the integral:

∫6x2 dx = 6∫x2 dx

And now use the Power Rule on x2:

= 6 x33 + C

Simplify:

= 2x3 + C

### Example: What is ∫(cos x + x) dx ?

Use the Sum Rule:

∫(cos x + x) dx = ∫cos x dx + ∫x dx

Work out the integral of each (using table above):

= sin x + x2/2 + C

### Example: What is ∫(ew − 3) dw ?

Use the Difference Rule:

∫(ew − 3) dw =∫ew dw − ∫3 dw

Then work out the integral of each (using table above):

= ew − 3w + C

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### Example: What is ∫(8z + 4z3 − 6z2) dz ?

Use the Sum and Difference Rule:

∫(8z + 4z3 − 6z2) dz =∫8z dz + ∫4z3 dz − ∫6z2 dz

Constant Multiplication:

= 8∫z dz + 4∫z3 dz − 6∫z2 dz

Power Rule:

= 8z2/2 + 4z4/4 − 6z3/3 + C

Simplify:

= 4z2 + z4 − 2z3 + C

### Integration by Parts

See Integration by Parts

### Substitution Rule

See Integration by Substitution