Problem 24 - Essential Calculus

Question

Find the integral. $$ \int(x+1) \sqrt{x^{2}+2 x+2} d x $$

Step-by-Step Solution

Answer

The integral of \((x+1) \sqrt{x^{2}+2x+2}\) with respect to \(x\) is \(\frac{1}{3}(x^2 + 2x + 2)^{3/2} + C.\)

Step 1: Substitute

Let \(u = x^2 + 2x + 2\). Calculate the derivative \(du = (2x+2)dx = 2(x+1)dx\). From here, we can express \(dx = du/(2(x+1))\). Replace \(x+1\) and \(dx\) in the integral, resulting in the following formula: \(\int u^{1/2} * du/2 = 0.5 \int u^{1/2} du.\)

Step 2: Integrate

Now perform the integration of \(0.5 \int u^{1/2} du\), which results in \( \frac{1}{3} u^{3/2}\)

Step 3: Back Substitute

Replace \(u\) with \(x^2 + 2x + 2\), resulting in \(\frac{1}{3}(x^2 + 2x + 2)^{3/2}.\) Add the constant of integration, \(C\).