3.3 Q-9

Question Statement

Evaluate the following integral:

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

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Background and Explanation

To solve this problem, we need to recognize a standard integral form that can be simplified using a trigonometric substitution. The key concepts involved include:

• Trigonometric substitution: Using the identity $1 + \tan^2 \theta = \sec^2 \theta$ to simplify the integrand.
• Basic trigonometric identities: Such as $\sec^2 \theta = 1 + \tan^2 \theta$ and $\cos \theta = \frac{1}{\sec \theta}$.

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Solution

Let’s solve this step by step:

1. Substitute using trigonometric identities: We choose the substitution: $x = \tan \theta$ Then, we differentiate to get: $dx = \sec^2 \theta , d\theta$

2. Rewrite the integral: Substituting $x = \tan \theta$ into the original integral, we get:

$$\int \frac{dx}{(1 + x^2)^{\frac{3}{2}}} = \int \frac{\sec^2 \theta , d\theta}{(1 + \tan^2 \theta)^{\frac{3}{2}}}$$

3. Simplify using a standard trigonometric identity: Recall that: $1 + \tan^2 \theta = \sec^2 \theta$ Thus, the integral becomes:

$$= \int \frac{\sec^2 \theta , d\theta}{(\sec^2 \theta)^{\frac{3}{2}}} = \int \frac{\sec^2 \theta , d\theta}{\sec^3 \theta}$$

4. Simplify further: We can now cancel out one factor of $\sec \theta$ from the numerator and denominator:

$$= \int \frac{d\theta}{\sec \theta} = \int \cos \theta , d\theta$$

5. Integrate: The integral of $\cos \theta$ is straightforward:

$$\int \cos \theta , d\theta = \sin \theta + C$$

6. Substitute back in terms of x: Recall that $x = \tan \theta$, so we can express $\sin \theta$ in terms of $x$:

$$\sin \theta = \sqrt{1 - \frac{1}{1 + x^2}} = \frac{x}{\sqrt{1 + x^2}}$$

7. Final result: Therefore, the integral evaluates to:

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

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Key Formulas or Methods Used

• Trigonometric substitution: $x = \tan \theta$
• Basic trigonometric identities:
  • $1 + \tan^2 \theta = \sec^2 \theta$
  • $\sec \theta = \frac{1}{\cos \theta}$

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Summary of Steps

1. Use the substitution $x = \tan \theta$ to simplify the integrand.
2. Simplify the integral using the identity $1 + \tan^2 \theta = \sec^2 \theta$.
3. Express the integral in terms of $\sec \theta$ and simplify.
4. Integrate $\cos \theta$ to get $\sin \theta + C$.
5. Substitute back to express the result in terms of $x$.
6. Final result: $\frac{x}{\sqrt{1 + x^2}} + C$.