Question

find the indefinite integral and check the result by differentiating.

$$ int x sqrt { x + 4 } d x $$

determine an appropriate substitution to simplify the integrand. choose the correct answer below.

a. $$ u = x - 4 $$
b. $$ u = x + 4 $$
c. $$ u = sqrt { x } $$
d. $$ u = sqrt { x + 4 } $$

Explanation:

Step1: Analyze the integrand

The integrand is \(x\sqrt{x + 4}\). If we let \(u=x + 4\), then \(x=u - 4\) and \(dx=du\). This substitution will simplify the integrand.

Step2: Check other options

• Option A: If \(u=x-4\), then \(x=u + 4\) and \(\sqrt{x + 4}=\sqrt{u+8}\), which does not simplify the integrand well.
• Option C: If \(u = \sqrt{x}\), then \(x=u^{2}\) and \(\sqrt{x + 4}=\sqrt{u^{2}+4}\), \(dx = 2udu\). The integrand becomes \(u^{2}\cdot\sqrt{u^{2}+4}\cdot2u\), which is more complex.
• Option D: If \(u=\sqrt{x + 4}\), then \(x=u^{2}-4\) and \(dx = 2udu\). The integrand becomes \((u^{2}-4)\cdot u\cdot2udu\), which is more complex than with \(u=x + 4\).

Answer:

B. \(u=x + 4\)