∫ x tan −1 x dx

To solve the integral ∫ x tan⁻¹(x) dx, we can use integration by parts. This method is useful when dealing with products of functions. Here’s how to approach it:

Step 1: Identify Parts

We choose:

• u = tan⁻¹(x) (which we will differentiate)
• dv = x dx (which we will integrate)

Step 2: Differentiate and Integrate

Now, we find the derivatives and integrals:

• du = (1 / (1 + x²)) dx
• v = (1/2)x²

Step 3: Apply Integration by Parts Formula

The integration by parts formula is:

∫ u dv = uv - ∫ v du

Substituting our values, we get:

∫ x tan⁻¹(x) dx = (1/2)x² tan⁻¹(x) - ∫ (1/2)x² (1 / (1 + x²)) dx

Step 4: Simplify the Remaining Integral

The remaining integral can be simplified:

∫ (1/2)x² / (1 + x²) dx = (1/2) ∫ (x² / (1 + x²)) dx

This can be rewritten as:

(1/2) ∫ (1 - 1/(1 + x²)) dx

Step 5: Solve the Integral

Now we can integrate:

• The integral of 1 dx is x.
• The integral of 1/(1 + x²) dx is tan⁻¹(x).

Putting it all together, we have:

(1/2)(x - tan⁻¹(x)) + C

Final Result

Combining everything, the solution to the integral is:

∫ x tan⁻¹(x) dx = (1/2)x² tan⁻¹(x) - (1/2)(x - tan⁻¹(x)) + C

Where C is the constant of integration. This gives you the complete solution to the integral.