To evaluate the integral

$I = \int_{0}^{1} \frac{(\arctan x)^2}{1+x^2} \, dx,$

we proceed as follows:

Step 1: Recognize a substitution

The derivative of $\arctan x$ is $\frac{1}{1+x^2}$. Therefore, we substitute: $u = \arctan x, \quad \text{so that} \quad du = \frac{1}{1+x^2} dx.$

When $x = 0$, $u = \arctan 0 = 0$.

When $x = 1$, $u = \arctan 1 = \frac{\pi}{4}$.

The integral becomes: $I = \int_{0}^{\pi/4} u^2 \, du.$

Step 2: Solve the integral

To compute $\int u^2 \, du$, use the power rule for integration: $\int u^n \, du = \frac{u^{n+1}}{n+1} + C.$ Here, $n = 2$: $\int u^2 \, du = \frac{u^3}{3}.$

Thus: $I = \left[\frac{u^3}{3}\right]_{0}^{\pi/4}.$

Step 3: Apply the limits

Evaluate the definite integral: $I = \frac{(\pi/4)^3}{3} - \frac{(0)^3}{3}.$ $I = \frac{\pi^3}{3 \cdot 4^3} = \frac{\pi^3}{192}.$

Final Answer:

$I = \frac{\pi^3}{192}.$

Would you like a detailed breakdown of any specific step?

Related Questions:

1. What is the general procedure for integrating functions involving $\arctan x$?
2. How does substitution simplify this integral?
3. Can you derive the formula for $\int u^n \, du$?
4. What other techniques could you use to evaluate this integral?
5. What are the properties of $\arctan x$ and its derivatives?

Tip:

When integrating expressions involving $\arctan x$, look for substitutions based on the derivative of $\arctan x$, which simplifies the process significantly!