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Integration Rules


Integration Integration  can be used to find areas, volumes, central points and many useful things. But it is often used to find the ...


Integration can be used to find areas, volumes, central points and many useful things. But it is often used to find the area underneath the graph of a function like this:
integral area

The integral of many functions are well known, and there are useful rules to work out the integral of more complicated functions, many of which are shown here.
There are examples below to help you.
Common FunctionsFunctionIntegral
Constanta dxax + C
Variablex dxx2/2 + C
Squarex2 dxx3/3 + C
Reciprocal(1/x) dxln|x| + C
Exponentialex 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 constantcf(x) dxcf(x) dx
Power Rule (n≠-1)xn dxxn+1/(n+1) + C
Sum Rule(f + g) dxf dx + g dx
Difference Rule(f - g) dxf 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

Power Rule

Example: What is x3 dx ?

The question is asking "what is the integral of x?"
We can use the Power Rule, where n=3:
xn dx = xn+1/(n+1) + C
xdx = x4/4 + C

Example: What is √x dx ?

√x is also x0.5
We can use the Power Rule, where n=½:
xn dx = xn+1/(n+1) + C
x0.5 dx = x1.5/1.5 + C

Multiplication by constant

Example: What is 6x2 dx ?

We can move the 6 outside the integral:
6x2 dx = 6x2 dx
And now use the Power Rule on x2:
= 6 x3/3 + C
= 2x3 + C

Sum Rule

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

Difference Rule

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

Sum, Difference, Constant Multiplication And Power Rules

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:
= 8z dz + 4z3 dz − 6z2 dz
Power Rule:
= 8z2/2 + 4z4/4 − 6z3/3 + C
= 4z2 + z4 − 2z3 + C

Integration by Parts

Substitution Rule



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Ilm Gah: Integration Rules
Integration Rules
Ilm Gah
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