Friday, October 7, 2011

`int 3/(2sqrt(x)(1+x)) dx` Find the indefinite integral

For the given integral: `int 3/(2sqrt(x)(1+x)) dx` , we may apply the basic integration property: `int c*f(x) dx = c int f(x) dx` .


`int 3/(2sqrt(x)(1+x)) dx = 3/2int 1/(sqrt(x)(1+x)) dx` .



For the integral part, we apply u-substitution by letting:


`u = sqrt(x) `


We square both sides to get: `u^2 = x` .


Then apply implicit differentiation, we take the derivative on both sides with respect to x as:


`2u du =dx` .


Plug-in `dx= 2u du` , `u =sqrt(x) ` and` x= u^2` in the integral:


`3/2int 1/(sqrt(x)(1+x)) dx =3/2int 1/(u(1+u^2)) (2u du)`


Simplify by cancelling out u and 2 from top and bottom:


`3/2int 1/(u(1+u^2)) (2u du) =3 int 1/(1+u^2) du`


The integral part resembles the basic integration formula  for inverse tangent:


`int 1/(1+u^2) du = arctan (u) +C`


 then, 


`3 int 1/(1+u^2)  du = 3 * arctan(u) +C`


Express in terms x by plug-in  `u =sqrt(x)` :


`3 arctan(u) +C =3 arctan(sqrt(x)) +C`


Final answer:


`int 3/(2sqrt(x)(1+x)) dx = 3arctan(sqrt(x))+C`

No comments:

Post a Comment

Thomas Jefferson's election in 1800 is sometimes called the Revolution of 1800. Why could it be described in this way?

Thomas Jefferson’s election in 1800 can be called the “Revolution of 1800” because it was the first time in America’s short history that pow...