Integrate √x-1 dx

\(\int \sqrt{x-1}dx\)

For this, you do a simple u-substitution.

Set \(x-1\) equal to \(u\), giving you the following expression:

\(\int \sqrt{u}dx\)

With u-substitution, you then take the derivative of \(u=x-1\):

\(du=dx\)

This means we replace the \(dx\) part of the original expression with \(du\) as well:

\(\int \sqrt{u}du\)

Now the expression is easier to integrate:

\(\int \sqrt{u}du\)

\(\int u^{\frac{1}{2}}du\)

Add 1 to the exponent, and divide the expression by its sum of 3/2:

\(\frac{\int \left(u^{\frac{1}{2}+1}\right)}{\frac{1}{2}+1}du\)

Integration complete:

\(\frac{u^{\frac{3}{2}}}{\frac{3}{2}}+C\)

Now, we can't use \(u\) anymore, we need to replace it back with \(x-1\):

\(\frac{2\left(x-1\right)^{\frac{3}{2}}}{3}+C\)