Integration Rules

To find many useful things like area, volumes, central points etc we use Integration. 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.

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) + ∫ g d (a)

Difference Rule

∫ (f – g) da

∫f d(a) – ∫ g d (a)

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

  • Integration by parts
  • Substitution

Integration Rules Examples

Question 1: What is ∫ 8 a 3 da?

Solution: We can take 8 out of integral,

∫ 8 a 3 da = 8 ∫ a 3 da

= 8 a4 / 4 + C

= 2 a4 + C

Question 2: What is ∫ 4 a 3 da?

Solution: We can take 8 out of integral,

∫ 4 a 3 da = 4 ∫ a 3 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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