Evaluate `int (xtan^(-1)x)/(1+x^2)^(3/2) dx`

Answer

To evaluate the integral

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

we will use the substitution \( t = \tan^{-1} x \).

### Step 1: Substitution
Let \( t = \tan^{-1} x \). Then, we have:

\[ x = \tan t. \]

### Step 2: Differentiate
Differentiating both sides with respect to \( x \):

\[ \frac{dx}{dt} = \sec^2 t \implies dx = \sec^2 t \, dt. \]

### Step 3: Substitute in the Integral
Now, substituting \( x \) and \( dx \) into the integral, we get:

\[ I = \int \frac{\tan t \cdot t}{(1 + \tan^2 t)^{3/2}} \cdot \sec^2 t \, dt. \]

Using the identity \( 1 + \tan^2 t = \sec^2 t \), we can simplify:

\[ I = \int \frac{\tan t \cdot t}{(\sec^2 t)^{3/2}} \cdot \sec^2 t \, dt = \int \frac{\tan t \cdot t}{\sec^3 t} \cdot \sec^2 t \, dt = \int t \cdot \frac{\tan t}{\sec t} \, dt. \]

### Step 4: Simplify Further
Since \( \frac{\tan t}{\sec t} = \sin t \), we rewrite the integral:

\[ I = \int t \sin t \, dt. \]

### Step 5: Integration by Parts
Now we will use integration by parts. Let:

- \( u = t \) \(\Rightarrow du = dt\)
- \( dv = \sin t \, dt \) \(\Rightarrow v = -\cos t\)

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \):

\[ I = -t \cos t - \int -\cos t \, dt = -t \cos t + \int \cos t \, dt. \]

### Step 6: Integrate
The integral of \( \cos t \) is:

\[ \int \cos t \, dt = \sin t. \]

Thus, we have:

\[ I = -t \cos t + \sin t + C, \]

where \( C \) is the constant of integration.

### Step 7: Back Substitute
Now we substitute back \( t = \tan^{-1} x \):

\[ I = -\tan^{-1} x \cdot \cos(\tan^{-1} x) + \sin(\tan^{-1} x) + C. \]

### Step 8: Find \( \sin(\tan^{-1} x) \) and \( \cos(\tan^{-1} x) \)
Using the right triangle definitions:

- \( \sin(\tan^{-1} x) = \frac{x}{\sqrt{1 + x^2}} \)
- \( \cos(\tan^{-1} x) = \frac{1}{\sqrt{1 + x^2}} \)

Substituting these into our expression gives:

\[ I = -\tan^{-1} x \cdot \frac{1}{\sqrt{1 + x^2}} + \frac{x}{\sqrt{1 + x^2}} + C. \]

### Final Answer
Thus, the evaluated integral is:

\[ I = \frac{x - \tan^{-1} x}{\sqrt{1 + x^2}} + C. \]