Integration have the right to be provided to uncover areas, volumes, main points and many valuable things. The is regularly used to find the area underneath the graph that a role and the x-axis.

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The an initial rule to recognize is that integrals and also derivatives room opposites!

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Sometimes we can work the end an integral,because we recognize a corresponding derivative.

Integration Rules

Here are the most valuable rules, with instances below:


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 preeminence (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 through PartsSee Integration through Parts
Substitution RuleSee Integration by Substitution

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

From the table over it is provided as gift −cos(x) + C

It is written as:

∫sin(x) dx = −cos(x) + C


Example: what is the integral that 1/x ?

From the table over it is detailed as gift ln|x| + C

It is created as:

∫(1/x) dx = ln|x| + C

The vertical bars || either next of x average absolute value, because we don"t want to give an unfavorable values to the organic logarithm function ln.


Example: What is ∫x3 dx ?

The question is questioning "what is the integral that x3 ?"

We have the right to use the power Rule, whereby 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, wherein 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 exterior the integral:

∫6x2 dx = 6∫x2 dx

And now use the Power dominion 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 the end the integral of each (using table above):

= sin x + x2/2 + C


Example: What is ∫(ew − 3) dw ?

Use the distinction Rule:

∫(ew − 3) dw =∫ew dw − ∫3 dw

Then work out the integral of every (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 through Substitution

Final Advice

Get plenty of practiceDon"t forget the dx (or dz, etc)Don"t forget the + C