3.3 Q-11

Question Statement

Evaluate the following integral:

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

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Background and Explanation

This problem involves the integration of a rational function with a square root. The key to simplifying this integral is rationalizing the expression to convert the square roots into a more manageable form. We will use substitution and integration formulas to solve it. A helpful concept here is recognizing the appearance of inverse trigonometric functions like $\sin^{-1} x$ during the process.

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Solution

Let’s solve the integral step by step:

1. Express the integral: The given integral is:

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

2. Rationalize the expression: To simplify the square roots, multiply both the numerator and denominator by $\sqrt{1 + x}$:

$$= \int \frac{\sqrt{1 + x}}{\sqrt{1 - x}} \times \frac{\sqrt{1 + x}}{\sqrt{1 + x}} , dx$$

3. Simplify: After multiplying, the integral becomes:

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

This simplifies to:

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

4. Split the integral: Now, split the integral into two parts:

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

5. Solve the first part: The first integral is a standard integral:

$$\int \frac{1}{\sqrt{1 - x^2}} , dx = \sin^{-1} x$$

6. Solve the second part: The second part requires a substitution. Let’s use $u = 1 - x^2$, so $du = -2x , dx$. This gives us:

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

Simplifying the constants and solving the integral, we get:

$$-\frac{1}{2} \left( 1 - x^2 \right)^{\frac{1}{2}} + C$$

7. Combine results: Now, combine both parts of the solution:

$$\sin^{-1} x - \sqrt{1 - x^2} + C$$

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Key Formulas or Methods Used

• Standard Integrals:
  • $\int \frac{1}{\sqrt{1 - x^2}} , dx = \sin^{-1} x$
• Rationalization:
  • Multiply both the numerator and denominator by $\sqrt{1 + x}$ to simplify the expression.
• Substitution:
  • Used substitution to simplify the second integral.

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Summary of Steps

1. Rationalize the given expression by multiplying both numerator and denominator by $\sqrt{1 + x}$.
2. Simplify the resulting expression.
3. Split the integral into two parts: $\int \frac{1}{\sqrt{1 - x^2}} , dx$ and $\int \frac{x}{\sqrt{1 - x^2}} , dx$.
4. Solve the first part as $\sin^{-1} x$.
5. Solve the second part using substitution, resulting in $-\frac{1}{2} \sqrt{1 - x^2}$.
6. Combine the results: $\sin^{-1} x - \sqrt{1 - x^2} + C$.