3.3 Q-18

Question Statement

Evaluate the following integral:

$$\int \frac{x , dx}{x^4 + 2x^2 + 5}$$

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Background and Explanation

This integral involves a rational function with a quartic denominator. To solve it, we will use trigonometric substitution. Specifically, we will recognize the structure of the denominator and apply the substitution $x^2 + 1 = 2 \tan \theta$, which will transform the integral into a form that is easier to handle. Understanding trigonometric identities and the properties of $\sec^2 \theta$ and $\tan^2 \theta$ will be essential.

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Solution

Let’s break down the solution step by step:

1. Substitution: We begin by making the substitution $x^2 + 1 = 2 \tan \theta$, which simplifies the denominator. This substitution implies:

$$2x , dx = 2 \sec^2 \theta , d\theta \quad \text{or} \quad x , dx = \sec^2 \theta , d\theta$$

Substituting these into the integral, we get:

$$\int \frac{\sec^2 \theta , d\theta}{\left( 2 \tan \theta \right)^2 + 2^2}$$

2. Simplify the Expression: Now simplify the expression:

$$= \int \frac{\sec^2 \theta , d\theta}{4 (\tan^2 \theta + 1)}$$

Since $\tan^2 \theta + 1 = \sec^2 \theta$, we can replace $\tan^2 \theta + 1$ with $\sec^2 \theta$ to further simplify the integral:

$$= \frac{1}{4} \int \frac{\sec^2 \theta , d\theta}{\sec^2 \theta}$$

3. Final Simplification: The $\sec^2 \theta$ terms cancel out, leaving:

$$= \frac{1}{4} \int 1 , d\theta$$

4. Integrate: The integral of 1 with respect to $\theta$ is simply $\theta$. So, we get:

$$= \frac{1}{4} \theta + C$$

5. Substitute Back for $\theta$: Now, recall the substitution $x^2 + 1 = 2 \tan \theta$, so $\theta = \tan^{-1} \left( \frac{x^2 + 1}{2} \right)$. Substituting this back into the expression:

$$= \frac{1}{4} \tan^{-1} \left( \frac{x^2 + 1}{2} \right) + C$$

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Key Formulas or Methods Used

• Trigonometric Substitution: We used $x^2 + 1 = 2 \tan \theta$ to simplify the integral.
• Basic Integration: The integral of $1 , d\theta$ is $\theta$.
• Trigonometric Identity: $\sec^2 \theta = 1 + \tan^2 \theta$ was used to simplify the integrand.

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Summary of Steps

1. Substitute $x^2 + 1 = 2 \tan \theta$ to simplify the denominator.
2. Express $x , dx = \sec^2 \theta , d\theta$ for the differential.
3. Simplify the integral to $\frac{1}{4} \int 1 , d\theta$.
4. Integrate and obtain $\frac{1}{4} \theta + C$.
5. Substitute back $\theta = \tan^{-1} \left( \frac{x^2 + 1}{2} \right)$ to get the final result:

$$\frac{1}{4} \tan^{-1} \left( \frac{x^2 + 1}{2} \right) + C$$