lunes, 27 de junio de 2011

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The dissimilar leisure sabotages another antique underneath the trapped crossroad.

lunes, 9 de mayo de 2011

More Examples of Integral by Substitution

#1

$\int\frac{tan^{-1}x}{1+x^{2}}\;dx$

$u=tan^{-1}x$
$du=\frac{dx}{1+x^2}$
$\int u\;du=\frac{u^2}{2}+C$
$\frac{(tan^{-1}x)^{2}{}}{2}+C$

#2

$\int sinx*cosx\;dx$
$u= sinx$
$du= cosxdx$
$\int u\;du= \frac{1}{2} u^2+c$
$\frac{1}{2} sin^2x+c$

#3

$\int sin^2 3x\cos 3x\;dx$
$u= 3x$
$du= 3dx$
$\frac{1}{3}du= dx$
$\frac{1}{3}\int sin^2 u\cos u\;du$
$u= sin u$
$du= cos u\;du$
$\frac{1}{3}\int u^2\;du = \frac{1}{3}\;(\frac{1}{3}u^3)+c$
$\frac{1}{9}u^3+c$
$\frac{1}{9}sin^3\;3x+c$

#4

$\int e^x(1+e^x)^{10}\;dx$
$u= 1+e^x$
$du= e^x\;dx$
$\int u^{10}\;du$
$=\frac{u^{11}}{11}$
$=\frac{(1+e^x)^{11}}{11}+C$

#5

$\int \frac{(lnx)^2}{x}\;dx$
$u= lnx$
$du= \frac{1}{x}\;dx$
$\int u^2\;du$
$=\frac{(u)^3}{3}$
$=\frac{(lnx)^3}{3}+C$

#6

$$\int \frac{1}{2x-1}\;dx$$

$$u= 2x-1$$

$$du= 2dx$$

$$\frac{1}{2}du=dx$$


$$\frac{1}{2}\int \frac{1}{u}\;du$$

$$=\frac{1}{2}lnu+C$$

$$=\frac{1}{2}ln(2x-1)+C$$

Integral by Substitution Example #4

$\int sinx*cosx\;dx$

$u= sinx$

$du= cosx \; dx$

$ \int u\;du= \frac{1}{2} u^2+c $

$ \frac{1}{2} sin^2x+c $

Integral by Substitution Example #3

Calculate



SOLUTION
Let


To find the limits of integration we note that when,



and when



Therefore

Integral by Substitution Example #2

Calculate


SOLUTION
Let

then

therefore

viernes, 6 de mayo de 2011

Integration by Sustitution

Lets start by understanding that this method of integration comes from the rule chain method for example,

(1)

lets integrate both sides of equation 1






we can use the substitution

working with

(2)

then



we make the substitution on the integral and get,



by the fundamental theorem of calcules we get



we use equation (2) and get that



Example #1

Solve the integral



We choose



and



we change the integral to

My first post!

For this blog im going to use latex. You can learn a lot by using this page http://www.codecogs.com/latex/eqneditor.php

Hope that the problems and solutions I post help you get better understanding of school and university math.

Bye