Question

how will you know the usage of techniques of integration?

Answer 1

Integration = 100% guessing
you will get good at it by practice only

Answer 2

http://mathcs.holycross.edu/~rjones/Math126s09/Integration_handout.pdf

this is insightful

Answer 3

these may help in establishing peace in you :
1) we cannot integrate every funciton
2) there are only few known manual methods for integrating special functions like polynomials/rational/trig etc
3) we can integrate all polynomials using power rule and all rational funcitons using partial fractions, and we cannot integrate most of the other funcitons manually

for example :
\[\large \int e^{x^2} dx = \text{we don't know }\]

Answer 4

So you cannot approach integrals in the same as derivatives

Answer 5

ahh. okay.. eh when we will use the appropriate techniques of integration for us to solve the integration easily?

Answer 6

First of all, the power rule enables you to integrate ANY polynomial :

Technique 1 :
\[\large \int x^n dx = \dfrac{x^{n+1}}{n+1}+C\]

Answer 7

yeah.. i know that, :)

Answer 8

good :) next you just need to know \(\large u \) substitution
its mostly a guessing game, but u will get it after doing 2-3 example problems

Answer 9

*

Answer 10

lets try one example problem ?

Answer 11

sure. :)

Answer 12

see if you work below integral :

\[\large \int 2x e^{x^2}dx \]

Answer 13

i will use integration by parts. :)

Answer 14

avoid integration by parts by all means

Answer 15

always try u-substitution first

Answer 16

aww! i can't use that rather.. :(

Answer 17

steps in u substitution :
1) substitue \(u\) = something
2) find \(dx\) in terms of \(du\)
3) integrate the new function
4) substitute back the \(u\)

Answer 18

\[\large \int \color{red}{2x} e^{\color{red}{x^2}}dx\]

Answer 19

Notice that derivative of \(\large \color{Red}{x^2}\) is \(\large \color{Red}{2x}\)
so that should hind us to use u substitution

Answer 20

ahh. gets.

u = x^2
du = 2xdx
1/2 du = xdx


∫2xe^x^2dx
2∫xe^x^2dx
2∫1/2 e^u du

right?

Answer 21

substitute \(\large u = \color{Red}{x^2} \)
\(\large du =\color{Red}{2x} ~dx \)

so the integral becomes :
\[\large \int e^{u}du\]

Answer 22

yes.. i got that :)

Answer 23

and its like your best of dreams, what else u can expect to integrate other than an exponential function ?

Answer 24

coz integral of e^x is e^x itself,
the most easy function to integrate

Answer 25

\[\large \int e^{u}du\]

evaluates to :

\[\large e^{u} + C\]

Answer 26

substitute back \(u = x^2\)

Answer 27

the final answer is :

\[\large e^{x^2} + C\]

Answer 28

yeah.. i got the answer. :)

Answer 29

sorry i didn't let you do the problem haha!
il let u work next example

Answer 30

it's okay. :)

Answer 31

try to work below integral using a suitable u substitution :
\[\large \int \dfrac{\ln x}{x} dx = ?\]

Answer 32

u = ln x
du = dx/x

∫udu = u^2/2 + c = ln^2x / 2 + c

Answer 33

Excellent !

Answer 34

yehey! :)

Answer 35

looks you have mastered u-substitution already :)

Answer 36

do u have any specific integral u want us to work ?

Answer 37

am, i have... wait, i'll draw it.

Answer 38

okie