Animated Solution for Mathematics - Definite Integration: The value of the integral ${\int }_{-1}^2{\log }_e(x+\sqrt{x^2+1})dx$ is

When you first look at the integral

it is natural to feel a bit of hesitation. It looks complex, almost transcendental, and it does not fit the standard patterns you might be used to.

However, in JEE Advanced, the most intimidating problems often have the most elegant solutions. We are going to solve this using the powerful technique of Integration by Parts (IBP).

Why IBP?

Whenever you see a standalone logarithmic or inverse trigonometric function, your intuition should immediately jump to Integration by Parts. The formula

is our master key. We treat our integrand as $u\cdot 1$, where $u=\ln (x+\sqrt{x^2+1})$ and $dv=dx$.

By choosing $u$ this way, we are betting that its derivative will be simpler than the function itself. And spoiler alert: it is.

Let us differentiate $u=\ln (x+\sqrt{x^2+1})$. Using the chain rule, the derivative of $\ln (f(x))$ is $\frac{1}{f(x)}\cdot f^{\prime }(x)$.

So,

Look closely at the term inside the parenthesis. If we simplify $\frac{2x}{2\sqrt{x^2+1}}$ to $\frac{x}{\sqrt{x^2+1}}$ and then find a common denominator, the numerator becomes $\sqrt{x^2+1}+x$.

This is exactly the same as the denominator of our first term! They cancel out beautifully, leaving us with

This is the 'Aha!' moment. The complexity has vanished.

Now, we have $v=x$ and $du=\frac{1}{\sqrt{x^2+1}}dx$. Plugging these into our IBP formula, we get

The second integral, $\int \frac{x}{\sqrt{x^2+1}}dx$, is a classic substitution problem. Let $t=x^2+1$, which means $dt=2xdx$, or $xdx=\frac{1}{2}dt$.

This transforms the integral into

Now, we just need to be careful with our limits. Evaluating the first part at $x=2$ gives $2\ln (2+\sqrt{5})$, and at $x=-1$ gives $-\ln (-1+\sqrt{2})$.

Combining these with the evaluated second part, $\sqrt{5}-\sqrt{2}$, we arrive at our final answer:

This matches our fourth option perfectly. Remember, the key to these problems is not just memorizing formulas, but understanding the flow of the logic. You have just navigated a complex integral by breaking it down into manageable, elegant steps.