Integrate arctan(1/t)/(1+t*t) from 0 to 1


To compute the integral of the function \( \frac{\arctan\left(\frac{1}{t}\right)}{1 + t^2} \) from 0 to 1, we proceed with a substitution and careful analysis.

The integral is given by:

\[
I = \int_0^1 \frac{\arctan\left(\frac{1}{t}\right)}{1 + t^2} \, dt
\]

### Step 1: Substitution

Let us make the substitution \( t = \frac{1}{u} \). This transforms the variable \( t \) in terms of \( u \).

For \( t = \frac{1}{u} \):
- \( dt = -\frac{1}{u^2} du \)
- When \( t = 0 \), \( u \to \infty \)
- When \( t = 1 \), \( u = 1 \)

Thus, the limits of integration become from \( \infty \) to 1, and the integral transforms as follows:

\[
I = \int_\infty^1 \frac{\arctan(u)}{1 + \frac{1}{u^2}} \cdot \left( -\frac{1}{u^2} \right) du
\]

Now simplify the denominator:

\[
1 + \frac{1}{u^2} = \frac{u^2 + 1}{u^2}
\]

Thus, the integral becomes:

\[
I = \int_1^\infty \frac{\arctan(u)}{u^2 + 1} \, du
\]

### Step 2: Symmetry of the Integral

Now we see that the transformed integral looks very similar to the original one. In fact, it can be shown that:

\[
\int_0^1 \frac{\arctan\left(\frac{1}{t}\right)}{1 + t^2} \, dt = \int_1^\infty \frac{\arctan(t)}{1 + t^2} \, dt
\]

Thus, the original integral can be expressed as:

\[
I = \frac{1}{2} \int_0^\infty \frac{\arctan(t)}{1 + t^2} \, dt
\]

### Step 3: Known Result

It is a known result that:

\[
\int_0^\infty \frac{\arctan(t)}{1 + t^2} \, dt = \frac{\pi}{2} \ln(2)
\]

Therefore:

\[
I = \frac{1}{2} \cdot \frac{\pi}{2} \ln(2) = \frac{\pi}{4} \ln(2)
\]

### Final Answer:

The value of the integral is:

\[
I = \frac{\pi}{4} \ln(2)
\]

Would you like more details on any part of this process, or have any questions?

Here are 5 related questions to explore further:
1. How do you compute the integral of \( \frac{\arctan(x)}{1 + x^2} \) from 0 to infinity?
2. What are other methods of integrating functions involving inverse trigonometric functions?
3. How does the substitution \( t = \frac{1}{u} \) help simplify integrals?
4. Can the symmetry of functions always help reduce the complexity of definite integrals?
5. What are some applications of integrals involving \( \arctan(x) \)?

**Tip**: Always check for symmetries in an integral's limits or form—this can simplify the process significantly.

Integrate arctan(1/t)/(1+t*t) from 0 to 1

Learn how to compute the definite integral ∫_0^1 (arctan(1/t) / (1 + t^2)) dt using substitution and symmetry. Explore the key concepts of inverse trigonometric functions and known results, concluding with the answer (π/4) ln(2). This problem is ideal for advanced calculus students.

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