Calculus Integration Problem

Mr. K  asks:

Integrate:
(1+x^{\frac{1}{2}})^{-1} dx

Solving:

For the integrand \frac{1}{\sqrt{x}+1} substitute u=\sqrt{x} and du=\frac{1}{(2\sqrt{x})}dx
=2\int{\frac{u}{u+1}}du
For the integrand \frac{u}{u+1}, do long division:
=2\int{(1-\frac{1}{u+1})}du
Integrate the sum term by term and factor out constants:
=2\int1du-2\int{\frac{1}{u+1}}du
For the integrand \frac{1}{u+1} substitute s=u+1 and ds=du
=2\int1du-2\int\frac{1}{s}ds
The integral of \frac{1}{s} is log(s)
The integral of 1 is u
=2u-2\log{(u+1)} + constant
Substitute back for u=\sqrt{x} ( =x^1/2)
=2\sqrt{x} -2\log{(\sqrt{x}+1)} + constant
🙂
Advertisements

Leave a Reply

Fill in your details below or click an icon to log in:

WordPress.com Logo

You are commenting using your WordPress.com account. Log Out / Change )

Twitter picture

You are commenting using your Twitter account. Log Out / Change )

Facebook photo

You are commenting using your Facebook account. Log Out / Change )

Google+ photo

You are commenting using your Google+ account. Log Out / Change )

Connecting to %s