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The substitution method is an essential technique for simplifying integrals. The goal is to transform a complicated integral into a simpler one. Here's a step-by-step breakdown of how it works:

Identify a part of the integrand that can be replaced with a single variable. This often makes the integral more manageable to work with.
In this case, we let \( u = x^2 + 1 \). This substitution simplifies the radical expression \( \sqrt{x^2 + 1} \).
Find the differential \( du \). Here, differentiating \( u \) with respect to \( x \) gives us \( du = 2x \, dx \). Notice that our integral contains \( 2x \, dx \), which aligns perfectly with \( du \).
By making this substitution, we can rewrite the entire integral in terms of \( u \). This often turns a complex integral into one that is straightforward to evaluate.

When using the substitution method, it is crucial to adjust the integration limits to match your new variable. This ensures that you evaluate the integral correctly within the new context:

Determine the new limits by substituting the original limits into your substitution equation.
For the integral \( \int_{0}^{1} \frac{2 x}{\sqrt{x^{2}+1}} \, dx \), the original limits are from 0 to 1.
Using the substitution \( u = x^2 + 1 \), we find that when \( x = 0 \), \( u = 0^2 + 1 = 1 \).
Similarly, when \( x = 1 \), \( u = 1^2 + 1 = 2 \).
Hence, the new integral becomes \( \int_{1}^{2} \frac{1}{\sqrt{u}} \, du \). Always replace the old limits with the new ones to properly evaluate the integral.

The final step in evaluating an integral is to perform the actual integration and then apply the limits to get a numerical answer:

Rewrite the integral in its simplified form using the new variable limits. In this example, \( \int_{1}^{2} \frac{1}{\sqrt{u}} \, du \) simplifies to \( \int_{1}^{2} u^{-1/2} \, du \).
Integrate the simplified function. The integral of \( u^{-1/2} \) is \( 2u^{1/2} \).
Now apply the new limits of integration. Substitute the upper limit first: \( 2[ \sqrt{2} ] \).
Then, substitute the lower limit: \( 2[ \sqrt{1} ] \).
Subtract these results: \( 2[ \sqrt{2} - \sqrt{1} ] \), which simplifies to \( 2(\sqrt{2} - 1) \).
This gives you the final answer: \( 2( \sqrt{2} - 1 ) \). Following these steps ensures you carry out the definite integral evaluation correctly and arrive at the correct result.