Question

int_{0}^{8} \frac{d x}{sqrt{1+x}}

Explanation:

Step1: Substitute variable

Let $u = 1 + x$, so $du = dx$. When $x=0$, $u=1$; $x=8$, $u=9$.

Step2: Rewrite integral

Integral becomes $\int_{1}^{9} u^{-1/2} du$.

Step3: Compute antiderivative

Antiderivative of $u^{-1/2}$ is $2u^{1/2}$.

Step4: Evaluate definite integral

$2(9^{1/2} - 1^{1/2}) = 2(3 - 1) = 4$.

Answer:

4