3.3 Q-3

Question Statement

Evaluate the integral:

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

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Background and Explanation

To solve this integral, we can use the technique of splitting the fraction into more manageable parts. This involves expressing the numerator in a way that simplifies the integral. In this case, we will rewrite $x^2$ as $(4 + x^2) - 4$ to split the expression into two terms. Recognizing the arctangent form for integrals will also be helpful.

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Solution

1. Rewrite the integrand:
   Start by splitting the numerator $x^2$ as follows:

$$x^2 = (4 + x^2) - 4$$

This gives us:

$$\frac{x^2}{4 + x^2} = \frac{(4 + x^2) - 4}{4 + x^2} = 1 - \frac{4}{4 + x^2}$$

So, the integral becomes:

$$\int \frac{x^2}{4 + x^2} , dx = \int \left(1 - \frac{4}{4 + x^2}\right) , dx$$

2. Split the integral:
   Now, we can break the integral into two separate parts:

$$\int \left(1 - \frac{4}{4 + x^2}\right) , dx = \int 1 , dx - 4 \int \frac{1}{4 + x^2} , dx$$

3. Solve the integrals:
   • The first integral is straightforward:

$$\int 1 , dx = x$$

• The second integral is a standard arctangent integral. We recognize that:

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

Substituting this back into our expression:

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

Simplifying:

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

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Key Formulas or Methods Used

• Splitting the Fraction: The technique of breaking the integrand into simpler parts, i.e., rewriting $x^2$ as $(4 + x^2) - 4$.
• Standard Integral of Arctangent: The formula:

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

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Summary of Steps

1. Rewrite the integrand:
   $\frac{x^2}{4 + x^2} = 1 - \frac{4}{4 + x^2}$

2. Split the integral into two parts:
   $\int 1 , dx - 4 \int \frac{1}{4 + x^2} , dx$

3. Solve the integrals:

   • $\int 1 , dx = x$
   • $\int \frac{1}{4 + x^2} , dx = \frac{1}{2} \tan^{-1}\left(\frac{x}{2}\right)$

4. Combine the results:
   $x - 2 \tan^{-1}\left(\frac{x}{2}\right) + C$