Question

∫₀⁸ (dx)/(√(1 + x))

Explanation:

Step1: Choose substitution

Let \( u = 1 + x \), then \( du = dx \). When \( x = 0 \), \( u = 1 \); when \( x = 8 \), \( u = 9 \).

Step2: Rewrite integral

The integral becomes \( \int_{1}^{9} \frac{du}{\sqrt{u}} \).

Step3: Integrate

Integrate \( u^{-\frac{1}{2}} \), the antiderivative is \( 2u^{\frac{1}{2}} + C \).

Step4: Evaluate definite integral

Evaluate from 1 to 9: \( 2(9^{\frac{1}{2}} - 1^{\frac{1}{2}}) = 2(3 - 1) = 4 \).

Answer:

4