3.3 Q-20

Question Statement

Evaluate the following integral:

$$\int \frac{x+2}{\sqrt{x+3}} , dx$$

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Background and Explanation

In this problem, we are tasked with evaluating an integral that involves a rational function with a square root. A useful technique here is to break down the expression in a way that allows us to integrate each part separately. We will manipulate the integrand into simpler terms and apply basic integration rules.

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Solution

Let’s go through the solution step by step:

1. Rewrite the Numerator: Start by rewriting the numerator $x + 2$ as $(x + 3) - 1$:

$$\int \frac{x+2}{\sqrt{x+3}} , dx = \int \frac{(x+3) - 1}{\sqrt{x+3}} , dx$$

2. Separate the Integral: Now, split the integral into two separate parts:

$$\int \frac{(x+3)}{\sqrt{x+3}} , dx - \int \frac{1}{\sqrt{x+3}} , dx$$

3. Simplify Each Part: For the first integral, $\frac{x+3}{\sqrt{x+3}} = \sqrt{x+3}$. So, we have:

$$\int \sqrt{x+3} , dx$$

For the second part, $\frac{1}{\sqrt{x+3}} = (x+3)^{-\frac{1}{2}}$. So, we have:

$$\int (x+3)^{-\frac{1}{2}} , dx$$

4. Integrate Each Part:
   • The first integral is $\int \sqrt{x+3} , dx$. To solve this, we use the power rule of integration. Recall that the power rule is $\int x^n , dx = \frac{x^{n+1}}{n+1}$. Here, $n = \frac{1}{2}$, so:

$$\int \sqrt{x+3} , dx = \frac{(x+3)^{\frac{3}{2}}}{\frac{3}{2}} = \frac{2}{3} (x+3)^{\frac{3}{2}}$$

• The second integral is $\int (x+3)^{-\frac{1}{2}} , dx$. Applying the same power rule, we get:

$$\int (x+3)^{-\frac{1}{2}} , dx = \frac{2}{1} (x+3)^{\frac{1}{2}} = 2 \sqrt{x+3}$$

5. Combine the Results: Now, subtract the results of the two integrals:

$$\frac{2}{3} (x+3)^{\frac{3}{2}} - 2 \sqrt{x+3} + C$$

This is the final result for the given integral.

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Key Formulas or Methods Used

• Power Rule of Integration:
  • For $\int x^n , dx = \frac{x^{n+1}}{n+1}$, where $n \neq -1$.
• Simplification: Breaking the numerator into two terms allowed us to apply the power rule to each part separately.

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Summary of Steps

1. Rewrite $x + 2$ as $(x + 3) - 1$.
2. Separate the integral into two parts:
   • $\int \sqrt{x+3} , dx$
   • $\int (x+3)^{-\frac{1}{2}} , dx$
3. Integrate each part using the power rule:
   • $\int \sqrt{x+3} , dx = \frac{2}{3} (x+3)^{\frac{3}{2}}$
   • $\int (x+3)^{-\frac{1}{2}} , dx = 2 \sqrt{x+3}$
4. Combine the results:

$$\frac{2}{3} (x+3)^{\frac{3}{2}} - 2 \sqrt{x+3} + C$$