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Definition of Integration:

  • The way toward finding a function, given its anti-derivative, is called anti-differentiation (or integration).
  • In the event that F'(x) = f(x), we say F(x) is a integral of f(x). In arithmetic, a necessary relegates numbers to capacities in a way that can displacement, area, volume, and different ideas that emerge by joining little information. Integration is one of the two principle operations of math, with its backwards, differentiation, being the other. 
  • Given a capacity f of a genuine variable x and an interval [a, b] of the genuine line, the definite integral

∫^ba f(x)dx

    is characterized casually as the marked zone of the region in the xy-plane that is limited by the chart of f, the x-axis and the vertical lines x = a and x = b. The region over the x-   pivot adds to the aggregate and that underneath the x-hub subtracts from the total.

Explanation of Integration:

  • The integral sign ∫ speaks integration. The image dx, called the differential of the variable x, demonstrates that the variable of mix is x. The capacity f(x) to be integrated is known as the integrand. The image dx is isolated from the integrand by a space (as appeared). On the off chance that a capacity has a fundamental, it is said to be necessary. 

Integration Rules:

  • 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.

Common Function

Function

Integral

Constant

∫a dx

ax + C

Variable

∫x dx

x2/2 + C

Square

∫x2 dx

x3/3 + C

Reciprocal

∫(1/x) dx

ln|x| + C

Exponential

∫ex dx

ex + C

 

∫ax dx

ax/ln(a) + C

 

∫ln(x) dx

x ln(x) − x + C

Trigonometry (x in radians)

∫cos(x) dx

sin(x) + C

 

∫sin(x) dx

-cos(x) + C

 

∫tan(x) dx

ln|sec x|+ C

 

∫sec2(x) dx

tan(x) + C

Rules

Function

Integral

Multiplication by constant

∫cf(x) dx

c∫f(x) dx

Power Rule (n≠-1)

∫xn dx

xn+1/(n+1) + C

Sum Rule

∫(f + g) dx

∫f dx + ∫g dx

Difference Rule

∫(f - g) dx

∫f dx - ∫g dx


Examples:

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

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+1/ (n+1) + C
∫x3 dx = x4/4 + C

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

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 x3/3 + C 

Simplify:

= 2x3 + C

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

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

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:

Integration by Parts is a special method of integration that is often useful when two functions are multiplied together, but is also helpful in other ways.

∫u v dx = u∫v dx −∫u' (∫v dx) dx

  • u is the function u(x)
  • v is the function v(x)
  • What is ∫x cos(x) dx ?

So now it is in the format ∫u v dx we can proceed:

Differentiate u: u' = x' = 1
Integrate v: ∫v dx = ∫cos(x) dx = sin(x)

Integration by Substitution:

Integration by Substitution" (also called "u-substitution") is a method to find an integral, but only when it can be set up in a special way.

  • ∫cos(x2) 2x dx

Now integrate:

∫cos(u) du = sin(u) + C
And finally put u=x2 back again:
sin(x2) + C

 

Download Free Integration Formula PDF:

 

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