Questions: Use the substitution formula to evaluate the integral. ∫ from 0 to 1 sqrt(x+9) dx

Solution Steps

Let \( u = x + 9 \). Then, the differential \( du = dx \). We also need to change the limits of integration: when \( x = 0 \), \( u = 9 \) and when \( x = 1 \), \( u = 10 \).

The integral can now be rewritten as: \[ \int_{0}^{1} \sqrt{x+9} \, dx = \int_{9}^{10} \sqrt{u} \, du \]

Using the power rule for integration, we find: \[ \int \sqrt{u} \, du = \frac{2}{3} u^{3/2} \] Thus, we evaluate: \[ \left[ \frac{2}{3} u^{3/2} \right]_{9}^{10} = \frac{2}{3} (10^{3/2}) - \frac{2}{3} (9^{3/2}) \] Calculating this gives: \[ \frac{2}{3} (10\sqrt{10} - 27) \] This simplifies to: \[ -18 + \frac{20\sqrt{10}}{3} \]

Final Answer