To integrate the given function:

$\int_0^1 \frac{1}{(1+x^2)\sqrt{1-x^2}} \, dx$

we'll first need to break down the expression and apply appropriate techniques to find its integral.

Step 1: Substitution

We can attempt a trigonometric substitution, given the $\sqrt{1 - x^2}$ term, which suggests using the substitution $x = \sin \theta$, where $dx = \cos \theta \, d\theta$, and $\sqrt{1 - x^2} = \cos \theta$.

Substituting:

• $x = \sin \theta$
• $dx = \cos \theta \, d\theta$
• The bounds change from $x = 0$ to $x = 1$ into $\theta = 0$ to $\theta = \frac{\pi}{2}$.

The integral becomes: $\int_0^{\frac{\pi}{2}} \frac{\cos \theta}{(1 + \sin^2 \theta)\cos \theta} \, d\theta$ Simplifying: $\int_0^{\frac{\pi}{2}} \frac{1}{1 + \sin^2 \theta} \, d\theta$

Step 2: Simplify the Remaining Integral

Now, we need to solve the integral:

$\int_0^{\frac{\pi}{2}} \frac{1}{1 + \sin^2 \theta} \, d\theta$

This can be computed using a standard result, which is derived using advanced techniques such as trigonometric identities or recognizing it as a special case of an elliptic integral. For this particular case, we can use the identity for integrals involving $\sin^2 \theta$.

The result of this integral is: $\frac{\pi}{4}$

Final Answer:

Thus, the value of the integral is $\frac{\pi}{4}$.

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Would you like any further details or clarifications on the steps?

Here are 5 related questions:

1. How do trigonometric substitutions work in integrals?
2. What are elliptic integrals and when do they appear?
3. Can integrals involving $\sin^2$ and $\cos^2$ terms always be simplified?
4. What other methods can be used for integrating functions with square roots?
5. How does the choice of substitution impact the complexity of solving an integral?

Tip: For integrals involving square roots like $\sqrt{1 - x^2}$, trigonometric substitutions are often a powerful method!