Integral of arctan(x) - Step-by-Step Solution | ML for Beginners

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Solution

Answer:

Step-by-Step Solution

Step 1

Use integration by parts: let u = arctan(x), dv = dx

Step 2

Find du and v

Step 3

Apply integration by parts formula: ∫u dv = uv - ∫v du

Step 4

For the remaining integral, use substitution w = 1+x², dw = 2x dx

Step 5

Combine results

🤖 Machine Learning Application

Arctan functions are crucial in ML for normalizing angles and in certain activation functions. The arctan integral appears in calculating probability distributions and in gradient descent optimization for bounded outputs.

Frequently Asked Questions

What is the integral of arctan(x)?

The integral of arctan(x) is x\arctan(x) - \frac{1}{2}\ln(1+x^2) + C, where C is the constant of integration.

How do you solve this integral?

Follow the step-by-step solution above, which uses integration by parts.

Where is this used in machine learning?

Arctan functions are crucial in ML for normalizing angles and in certain activation functions. The arctan integral appears in calculating probability distributions and in gradient descent optimization for bounded outputs.