Integration Rules

Integration is used to find many useful things like area, volumes, central points, etc. It is mostly used to find the area covered by the graph of a function. To work out the integral of more complicated functions than just the known ones, we have some integration rules. These rules can be studied below. Apart from these rules, there are many integral formulas that substitute the integral form.

Basic Integration Rules

Common Functions Functions Integrals
 Constant ∫ x da xa + c
 Variable ∫a da a2/2 + C
Square ∫a2 da a3/2 + C
Reciprocal ∫1/a da In |a| + C
Exponential ∫ea da ea + C
∫at da at/In(a) + C
∫In (a) da a In a – a + C
Trigonometry (t in radians) ∫cos(a) da Sin a + C
∫sin (a) da -Cos a + C
∫sec2a da tan a + C
Rule Function Integral
Multiplication by constant ∫ cf(a)da C ∫ f(a) da
Power Rule ∫ an da (an+1 / n + 1) +C
Sum Rule ∫ (f+g) da ∫f d(a) + ∫ gd (a)
Difference Rule ∫ (f – g) da ∫f d(a) – ∫ gd (a)

Apart from the above-given rules, there are two more integration rules like:

Integration by parts 

It is a special kind of integration method when two functions are multiplied together. The rule for integration by parts is

∫ u v da = u∫ v da – ∫ u'(∫ v da)da

Where

  • u is the function of u(a)
  • v is the function of v(a)
  • u’ is the derivative of the function u(a)

Integration by Substitution

Integration by substitution is also known as “Reverse Chain Rule” or “u-substitution Method” to find an integral.

The first step in this method is to write the integral in the form:

∫ f(g(x))g'(x)dx

Now, we can do a substitution as follows:

g(x) = a and g'(a) = da

Now substitute the equivalent values in the above form:

∫ f(a) da

Once you integrate the above form, finally substitute the original values.

Integration Rules Examples

Question 1: What is ∫ 8 a3 da?

Solution: We can take 8 out of integral,

∫ 8 a3 da = 8 ∫ a3 da

= 8 a4 / 4 + C

= 2 a4 + C

Question 2: What is ∫ 4 a3 da?

Solution: We can take 8 out of integral,

∫ 4 a3 da = 4 ∫ a3 da

= 4 a4 / 4 + C

= 1 a4 + C

Question 3: What is ∫ Cos a + a da ?

Solution: ∫ Cos a + a da = ∫ Cos a da + ∫ a da

= sin a + a2 /2 + C

Question 3: What is ∫ Sin a + a da ?

Solution: ∫ Sin a + a da = ∫ Sin a da + ∫ a da

= – Cos a + a2 /2 + C

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