Find the consumer surplus, given supply and demand equations > Find the Consumer Surplus, given the demand and supply equations $$ D(x)=\frac{405}{\sqrt{x}} $$ $$ S(x)=5\sqrt{x} $$ The equilibrium point is $(81,45)$. I know the formula for consumer surplus, but I am stuck on finding the integral of $405/\sqrt{x}$.

For context, I grabbed this picture from Wikipedia

!surplus

The red area is the integral of $D(x) - 45$ from $0$ to $81$. Namely, $$ \int_0^{81} \left(\frac{405}{\sqrt{x}}-45\right)\,dx $$ To integrate, write $405/\sqrt{x}$ as $x^{-1/2}$ and use the formula $\int x^a = x^{a+1}/(a+1)$.

To check the answer, you can use Wolfram Alpha: it's $3645$.