## Indefinite Integration

After a long series on differentiation and ‘Application of derivatives’,** Online IB Tutors** will now discuss Indefinite Integration. It consists of two different words indefinite and integration by **IB Maths Tutors .**

First of all, we shall learn about Integration.

** **Integration is the reverse process of differentiation so we can also call it as antiderivative. There is one more name for it, that is Primitive.

If f & g are functions of x such that g'(x) = f(x) then the function g is called a Primitive Or Antiderivative Or Integral of f(x) w.r.t. x and is written symbolically as:-

If

then here c is just an arbitrary constant. Value of c is not definite that’s why we call it **Indefinite Integration.**

**Techniques Of Integration-:**

There are a few important techniques used to solve problems based on an integration

**(i)** **Substitution or Change of Independent Variable- **If the derivative of a function is given in the question, then we should use the method of substitution to integrate that question.

**Example-1** Find integration of this problem

Ans- Here derivative of is 2x that is given in the question so we can substitute by some other variable. let =t

If we differentiate both sides

2x.dx=dt

so = +c

**(ii) Integration by part-** If we are given the product of two functions such that we are not able to use the method of substitution to integrate it, then we use Integration by parts. Suppose u and v are two functions then-

Note-: While using integration by parts, choose u & v such that we can easily apply above formula and reduce the given function from a product of two functions into a function that can be easily integrated. For this, we choose u in the order of ILATE. Here

I inverse function

L Logarithmic Functions

A Algebraic Functions

T Trigonometric functions

E Exponential function

Some people use the above formula in a different way

they choose F(x) in this order: LIPET

logs, Inverse, Polynomial, exponential, trigonometric

**(iii) Integration by Partial Fractions- **Partial fraction is a long and different topic. We use it in integration to simplify some complex fraction. I have attached a whole module on this topic at the end of the post

**(iv) When the Power of Numerator is More Than the Power of Denominator- **In this case, we first divide the numerator by denominator to make it a pure fraction, then we can use the partial fraction to simplify and integrate it.

**Integrals Of Some Special Type-**

Here we have some special types of functions and tricks to integrate them.

(i) or in these cases, we let f(x)=t

(ii) , in all these cases we convert

() into a perfect square

(iii) , in these cases we Express:-

px + q = A (differential co-efficient of denominator) + B

(iv)

(v)

(vi)

(vii) in this case, we take common and put = t

(ix) in this case, we take common and put = t If we

If we take care of all these rules properly, then we can solve all the problems of definite integration.

Here I am attaching a full module on Integration by Partial fraction

A pdf containing almost all formulas of indefinite integration

A pdf of a few questions based on the integration

### In the next post, I shall discuss Definite integration

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