Server Not Reachable. *This may be due to your internet connection or the nubtrek server is offline.*

Thought-Process to Discover Knowledge

Welcome to **nub****trek**.

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

Books and other education websites provide "matter-of-fact" knowledge. Instead, nubtrek provides a thought-process to discover knowledge.

In each of the topic, the outline of the thought-process, for that topic, is provided for learners and educators.

Read in the blogs more about the unique learning experience at nubtrek.continue

Welcome to **nub****trek**.

The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

**Just keep tapping** (or clicking) on the content to continue in the trail and learn. continue

The content is presented in small-focused learning units to enable you to

think,

figure-out, &

learn.

To make best use of nubtrek, understand what is available.

nubtrek is designed to explain mathematics and science for young readers. Every topic consists of four sections.

nub,

trek,

jogger,

exercise.

This captures the small-core of concept in simple-plain English. The objective is to make the learner to think about. continue

Trekking is bit hard, requiring one to sweat and exert. The benefits of taking the steps are awesome. In the trek, concepts are explained with exploratory questions and your thinking process is honed step by step. continue

This captures the essence of learning and helps one to review at a later point. The reference is available in pdf document too. This is designed to be viewed in a smart-phone screen. continue

This part does not have much content as of now. Over time, when resources are available, this section will have curated and exam-prep focused questions to test your knowledge. continue

Voice

Voice

Home

Integration by Substitution

When `f(x) = h(g(x)) g′(x)`

then,

`int f(x)dx = int h(y)dy`

where `y=g(x)`

and `dy = g′(x)dx` and so

» `int f(x)dx ``= [int h(y)dy]|_(y=g(x)) + c`

*plain and simple summary*

nub

*plain and simple summary*

nub

dummy

Integration by substitution : `int h(g(x)) g′(x) dx = [int h(y)dy]|_(y=g(x))`

*simple steps to build the foundation*

trek

*simple steps to build the foundation*

trek

You are learning the free content, however do shake hands with a coffee to show appreciation.

*To stop this message from appearing, please choose an option and make a payment.*

In this page, integration by substitution is explained with examples.

Starting on learning "". In this page, integration by substitution is explained with examples.

Consider the integration `int 2x sin x^2 dx`

We notice that `d/(dx) x^2 = 2x` is part of the integration.

Which of the following can be used to work out the integration?

- take `t=x^2` and substitute in the integrand
- noticing `d/(dx) x^2 = 2x` does not help

The answer is "take `t=x^2` and substitute in the integrand".

integration `int 2x sin x^2 dx`

Substitute `t=x^2` and `dt = 2xdx`.

`int 2x sin x^2 dx`

`= int sin t dt`

`= -cos t +c` *substitute `t=x^2`*

`= -cos x^2 + c`

Considering the integration `int f(x)dx` :

If it is observed that `f(x)` can be rewritten as

`f(x) = h[g(x)] g′(x)`

then,

`int f(x)dx = int h(y)dy`

where `y=g(x)`

and `dy = g′(x)dx`

This method is "integration by substitution".

Given a function in parametric form `y=rcos t` and `x=rsin t`, How to proceed with the integration `int ydx`?

- Substitute `y=r cos t` and `dx = d (r cos t)`. Then, proceed with variable of integration as `t`
- the function in parametric form cannot be integrated

The answer is 'Substitute `y=r cos t` and `dx = d (r cos t)`'

Given a function in parametric form `y=rcos t` and `x=rsin t`.

`int ydx`*substituting `y=rcos t` and `dx = d(r sin t)`. the `d()` means derivative of *

`= int r cost d(r sin t)`

`= int r cos t r cos t dt`

`= int r^2 cos^2 t dt`

This can be integrated.

*comprehensive information for quick review*

*Jogger*

*comprehensive information for quick review*

*Jogger*

**Integration by Substitution: ** When `f(x) = h(g(x)) g′(x)`

then,

`int f(x)dx = int h(y)dy`

where `y=g(x)`

and `dy = g′(x)dx` and so `int f(x)dx ``= [int h(y)dy]|_(y=g(x)) + c`

*practice questions to master the knowledge*

*Exercise*

*practice questions to master the knowledge*

*Exercise*

*Integrate `int tan x dx` *

*can be integrated with `tan x = - (cos x)′//cos x`**no known method to integrate*

*The answer is "can be integrated" `int tanx dx` `=int (- (cos x)′)/(cos x)` `=-log|cos x| + c`*

*Integrate `int cot x dx` *

*can be integrated with `cot x = (sin x)′//sin x`**no known method to integrate*

*The answer is "can be integrated" `int tanx dx` `=((sin x)′)/(sin x)` `=log|sin x| + c`*

*Integrate `int sec x dx` *

*can be integrated with `sec x ``= (sec x + tan x)′``//(sec x + tan x)`**no known method to integrate*

*The answer is "can be integrated" `int sec x dx` `=int ((sec x + tan x)′)/(sec x + tan x) dx` `=log|sec x + tan x|+c `*

*Integrate `int csc x dx` *

*can be integrated with `csc x ``= -(csc x + cot x)′``//(csc x + cot x)`**no known method to integrate*

*The answer is "can be integrated" `int csc x dx` `=int (-(csc x + cot x)′)/(csc x + cot x) dx` `=-log|cscx+ cotx| + c`*

*your progress details*

*Progress*

*About you*

*Progress*