Questions: In Exercises 11 to 14, find the indefinite integral using the substitution (x=2 tan theta). [ int frac2 x^2(4+x^2)^2 d x ]

Solution Steps

To solve the integral \(\int \frac{2 x^{2}}{\left(4+x^{2}\right)^{2}} d x\) using the substitution \(x = 2 \tan \theta\), follow these steps:

1. Substitute \(x = 2 \tan \theta\), which implies \(dx = 2 \sec^2 \theta \, d\theta\).
2. Replace \(x\) and \(dx\) in the integral with the expressions in terms of \(\theta\).
3. Simplify the integral using trigonometric identities.
4. Integrate with respect to \(\theta\).
5. Convert back to \(x\) using the inverse substitution \(\theta = \tan^{-1}(x/2)\).

We start with the integral

\[ \int \frac{2 x^{2}}{(4+x^{2})^{2}} \, dx \]

Using the substitution \(x = 2 \tan \theta\), we find that

\[ dx = 2 \sec^2 \theta \, d\theta \]

Substituting \(x\) and \(dx\) into the integral, we have:

\[ \int \frac{2 (2 \tan \theta)^{2}}{(4 + (2 \tan \theta)^{2})^{2}} \cdot 2 \sec^2 \theta \, d\theta \]

This simplifies to:

\[ \int \frac{16 \tan^{2} \theta \sec^{2} \theta}{(4 \tan^{2} \theta + 4)^{2}} \, d\theta \]

The expression simplifies further to:

\[ \int \sin^{2} \theta \, d\theta \]

Integrating \(\sin^{2} \theta\) gives:

\[ \frac{\theta}{2} - \frac{\sin \theta \cos \theta}{2} + C \]

Substituting back \(\theta = \tan^{-1} \left(\frac{x}{2}\right)\):

\[ \frac{1}{2} \tan^{-1} \left(\frac{x}{2}\right) - \frac{\sin \left(\tan^{-1} \left(\frac{x}{2}\right)\right) \cos \left(\tan^{-1} \left(\frac{x}{2}\right)\right)}{2} + C \]

Using the identity \(\sin(\tan^{-1}(u)) = \frac{u}{\sqrt{1+u^2}}\) and \(\cos(\tan^{-1}(u)) = \frac{1}{\sqrt{1+u^2}}\), we find:

\[ \sin \left(\tan^{-1} \left(\frac{x}{2}\right)\right) = \frac{\frac{x}{2}}{\sqrt{1+\left(\frac{x}{2}\right)^{2}}} = \frac{x}{\sqrt{x^{2}+4}} \]

\[ \cos \left(\tan^{-1} \left(\frac{x}{2}\right)\right) = \frac{2}{\sqrt{x^{2}+4}} \]

Thus, the integral evaluates to:

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

Final Answer