Math Snap

STEP 1

What is this asking?
We need to calculate a definite integral of a derivative.
Watch out!
Don't forget the Fundamental Theorem of Calculus!

STEP 2

1. Apply the Fundamental Theorem of Calculus
2. Evaluate the endpoints

STEP 3

The Fundamental Theorem of Calculus tells us that the integral of a derivative is just the original function evaluated at the bounds of integration.
It's like magic!
We're asked to calculate $$\int_{1}^{e} \frac{d}{dx} \left[ \frac{x \ln x}{1+x^2} \right] dx.$$ The Fundamental Theorem of Calculus says $$\int_a^b \frac{d}{dx} f(x) \, dx = f(b) - f(a).$$ In our case, $f(x) = \frac{x \ln x}{1+x^2}$, $a = 1$, and $b = e$.
So we need to evaluate $f(x)$ at $x = e$ and $x = 1$.

STEP 4

Let's evaluate $f(e)$:
$$f(e) = \frac{e \ln e}{1 + e^2}.$$ Since $\ln e = 1$, we get
$$f(e) = \frac{e \cdot 1}{1 + e^2} = \frac{e}{1 + e^2}.$$

STEP 5

Now let's evaluate $f(1)$:
$$f(1) = \frac{1 \cdot \ln 1}{1 + 1^2} = \frac{1 \cdot 0}{1 + 1} = \frac{0}{2} = 0.$$

STEP 6

So, our final integral is:
$$f(e) - f(1) = \frac{e}{1 + e^2} - 0 = \frac{e}{1 + e^2}.$$

The value of the definite integral is $\frac{e}{1 + e^2}$.