Animated Solution for Mathematics - Definite Integration: For $x>0$, let $f(x)={\int }_e^x\frac{\ln t}{1+t}dt$. Find the function $f(x)+f\left(\frac{1}{x}\right)$ and show that $f(e)+f\left(\frac{1}{e}\right)=\frac{1}{2}$.

Unlocking the Integral

Imagine you are standing before a complex integral, a function $f(x)={\int }_e^x\frac{\ln t}{1+t}dt$. It looks intimidating, but in mathematics, every fortress has a key.

Today, we are going to explore the beautiful symmetry hidden within this function and prove that $f(x)+f(1/x)=\frac{1}{2}(\ln x{)}^2$.

The Power of Substitution

Our journey begins with the function $f(1/x)$. To understand this, we perform a transformation by setting $t=1/u$, which implies $dt=-\frac{1}{u^2}du$.

When we change the variable, we must change the limits. As $t$ goes from $e$ to $1/x$, $u$ goes from $1/e$ to $x$. Substituting these into our integral, we get:

Using the logarithm property $\ln (1/u)=-\ln u$, the two negative signs—one from the log and one from the differential—cancel out. This leaves us with a much cleaner expression:

The Algebraic Decomposition

Now, we face the fraction $\frac{1}{u(1+u)}$. We use Partial Fractions to decompose this into $\frac{1}{u}-\frac{1}{1+u}$.

By distributing the $\ln u$ term, our integral splits into two distinct parts:

The first part is a standard integral. Since the derivative of $\ln u$ is $1/u$, the integral of $\frac{\ln u}{u}$ is simply $\frac{(\ln u{)}^2}{2}$.

Evaluating this from $1/e$ to $x$ gives us:

Since $\ln (1/e)=-1$, this simplifies beautifully to $\frac{(\ln x{)}^2}{2}-\frac{1}{2}$.

The Grand Synthesis

We now have the expression:

To find $f(x)+f(1/x)$, we add our original $f(x)={\int }_e^x\frac{\ln t}{1+t}dt$. Notice that the integral ${\int }_{1/e}^x\frac{\ln u}{1+u}du$ can be split at $e$:

The second part is exactly our $f(x)$. When we combine everything, the $f(x)$ terms cancel out, and the constant terms vanish, leaving us with the elegant result:

Finally, to show $f(e)+f(1/e)=1/2$, we simply plug in $x=e$. Since $\ln e=1$, the result is $\frac{1}{2}(1{)}^2=1/2$.