Answer
$2\ln\left(\dfrac {3}{2}\right)$
Work Step by Step
The question asks us to find the definite integral for the function:
$f(x)=\frac{1}{\sqrt x+x}$
Thus, we must begin by finding the definite integral first. Since the question asks us to solve for the integral using substitution, we should find a suitable candidate for the substitute term.
It should be realized that the function can be rewritten into the following:
$f(x)=\frac{1}{\sqrt x+x}=\frac{1}{\sqrt x(\sqrt x+1)}$
Now, through algebraic manipulations, the function has been rewritten into a form where the substitute term, $u$ can be used.
$u=\sqrt x +1$
$du=\frac{1}{2\sqrt x}dx$
Thus, it becomes very clear why all the manipulations above were done.
$f(x)=\frac{1}{\sqrt x(\sqrt x+1)}$
$\int \frac{1}{\sqrt x(\sqrt x+1)} dx=\int 2 \frac{1}{2\sqrt x(\sqrt x+1)} dx=2 \int \frac{1}{u} du= 2ln(u)+c= 2ln(\sqrt x +1)+c$
Thus, having reached the indefinite integral, we can use it to find the area under the curve using: $F(b)-F(a)$
$2ln(\sqrt 4 +1)-2ln(\sqrt 1 +1)=2ln(\frac {3}{2})$
