Substitution

[dear2009]

Hey everybody,



This substitution is confusing me

This is how I started
x = x^3 + 1
dx = 3x^2

2x / (x^3 + 1) dx
(1/3x^3) dx
(3x^-3) dx

this lead me to [x^ (-3 + 1)] / (-3 + 1)
1 / 2x^2 + C
= (x^-2/ -2) + C

so when I plug in the numbers
C - (2) / 2 (x^3 + 1)^2
C - (1/2) = 1
C = 3/2

y = 3/2 - [1/ 2(1)]
y = 3/2 - 1/2
y = 1
but that isnt the right answer. Can anybody show me the proper way

 

[royhaas]

Start off with \(\displaystyle u = (x^3+1), du=3x^2dx\).

 

[galactus]

When using substitution, look for something in the integral that when you take the derivative is something else in the integral.

See what I mean?. As Roy said, see the \(\displaystyle x^{3}+1\)?. Well, the derivative of that is \(\displaystyle 3x^{2}dx\). This is in the numerator and can be replaced

by the du substitution.

Here is another to illustrate the idea. \(\displaystyle \int\frac{5x^{4}}{x^{5}+10}dx\)

Looks icky, but is rather easy. Note the \(\displaystyle x^{5}\). Its derivative is \(\displaystyle 5x^{4}dx\). That is in the numerator and can be replaced by the du.

Get the idea now?. Don't try to paddle upstream. Go with the flow.