`int_(0)^(sqrt3)(xtan^(-1)x)/((1+x^(2))^(3//2))dx`

Options

A. `(4-pi)/(4sqrt3)`
B. `(pi-4)/(4sqrt2)`
C. `(3sqrt3-pi)/(6)`
D. `(4sqrt2)/(4-pi)`

Answer

To solve the integral

\[ I = \int_{0}^{\sqrt{3}} \frac{x \tan^{-1}(x)}{(1+x^2)^{3/2}} \, dx, \]

we will use the substitution \( x = \tan(\theta) \). This gives us:

1. **Substitution**:
- When \( x = 0 \), \( \theta = \tan^{-1}(0) = 0 \).
- When \( x = \sqrt{3} \), \( \theta = \tan^{-1}(\sqrt{3}) = \frac{\pi}{3} \).
- The differential \( dx = \sec^2(\theta) \, d\theta \).
- We also have \( 1 + x^2 = 1 + \tan^2(\theta) = \sec^2(\theta) \).

Thus, the integral transforms as follows:

\[ I = \int_{0}^{\frac{\pi}{3}} \frac{\tan(\theta) \theta \sec^2(\theta)}{(\sec^2(\theta))^{3/2}} \, d\theta. \]

2. **Simplifying the Integral**:
- The term \( (\sec^2(\theta))^{3/2} = \sec^3(\theta) \).
- Therefore, we can simplify the integral:

\[ I = \int_{0}^{\frac{\pi}{3}} \frac{\tan(\theta) \theta \sec^2(\theta)}{\sec^3(\theta)} \, d\theta = \int_{0}^{\frac{\pi}{3}} \theta \tan(\theta) \cos(\theta) \, d\theta. \]

3. **Using the Identity**:
- We know that \( \tan(\theta) = \frac{\sin(\theta)}{\cos(\theta)} \).
- Thus, we can rewrite the integral as:

\[ I = \int_{0}^{\frac{\pi}{3}} \theta \sin(\theta) \, d\theta. \]

4. **Integration by Parts**:
- Let \( u = \theta \) and \( dv = \sin(\theta) \, d\theta \).
- Then \( du = d\theta \) and \( v = -\cos(\theta) \).
- Applying integration by parts:

\[ I = -\theta \cos(\theta) \bigg|_{0}^{\frac{\pi}{3}} + \int_{0}^{\frac{\pi}{3}} \cos(\theta) \, d\theta. \]

5. **Evaluating the Boundaries**:
- The first term evaluates to:

\[ -\left( \frac{\pi}{3} \cdot \cos\left(\frac{\pi}{3}\right) - 0 \cdot \cos(0) \right) = -\left( \frac{\pi}{3} \cdot \frac{1}{2} \right) = -\frac{\pi}{6}. \]

6. **Evaluating the Remaining Integral**:
- The integral of \( \cos(\theta) \):

\[ \int_{0}^{\frac{\pi}{3}} \cos(\theta) \, d\theta = \sin(\theta) \bigg|_{0}^{\frac{\pi}{3}} = \sin\left(\frac{\pi}{3}\right) - \sin(0) = \frac{\sqrt{3}}{2} - 0 = \frac{\sqrt{3}}{2}. \]

7. **Combining Results**:
- Therefore, we have:

\[ I = -\frac{\pi}{6} + \frac{\sqrt{3}}{2}. \]

8. **Final Result**:
- To combine these, we can find a common denominator:

\[ I = -\frac{\pi}{6} + \frac{3\sqrt{3}}{6} = \frac{3\sqrt{3} - \pi}{6}. \]

Thus, the final answer is:

\[ \boxed{\frac{3\sqrt{3} - \pi}{6}}. \]