Question

use a change of variables to find the following \\(\int \frac{(\sqrt{x} + 2)^2}{5\sqrt{x}} \\, dx\\) \\(\int \frac{(\sqrt{x} + 2)^2}{5\sqrt{x}} \\, dx = \square\\).

Explanation:

Step1: Choose substitution

Let \( u = \sqrt{x} + 2 \). Then, find \( du \). First, differentiate \( u \) with respect to \( x \): \( \frac{du}{dx}=\frac{1}{2\sqrt{x}} \), so \( du=\frac{1}{2\sqrt{x}}dx \), and \( 2du = \frac{1}{\sqrt{x}}dx \).

Step2: Rewrite the integral

The integral is \( \int\frac{(\sqrt{x} + 2)^2}{5\sqrt{x}}dx=\frac{1}{5}\int(\sqrt{x} + 2)^2\cdot\frac{1}{\sqrt{x}}dx \). Substitute \( u \) and \( du \): since \( \frac{1}{\sqrt{x}}dx = 2du \) and \( (\sqrt{x}+2)=u \), the integral becomes \( \frac{1}{5}\int u^2\cdot2du=\frac{2}{5}\int u^2du \).

Step3: Integrate with respect to \( u \)

Integrate \( \frac{2}{5}\int u^2du \) using the power rule \( \int u^n du=\frac{u^{n + 1}}{n+1}+C \) (for \( n
eq - 1 \)). Here, \( n = 2 \), so \( \frac{2}{5}\cdot\frac{u^{3}}{3}+C=\frac{2u^{3}}{15}+C \).

Step4: Substitute back \( u \)

Replace \( u \) with \( \sqrt{x}+2 \): \( \frac{2(\sqrt{x}+2)^{3}}{15}+C \).

Answer:

\(\frac{2}{15}(\sqrt{x} + 2)^{3}+C\) (where \( C \) is the constant of integration)