Problem 10 - A Complete Resource Book in Mathematics for JEE Main

Question

$$ \int\left(\frac{\ln x-1}{(\ln x)^{2}+1}\right)^{2} d x \text { is equal to } $$ (A) \(\frac{x}{x^{2}+1}+c\) (B) \(\frac{\ln x}{(\ln x)^{2}+1}+c\) (C) \(\frac{x}{(\ln x)^{2}+1}+c\) (D) \(e^{x}\left(\frac{x}{x^{2}+1}\right)+c\)

Step-by-Step Solution

Answer

The integral evaluates to Option (B): \( \frac{\ln x}{(\ln x)^2+1}+c \).

Step 1: Substitute the variable

Let us use the substitution \( u = \ln x \), which implies \( du = \frac{1}{x} dx \). Therefore, the integral becomes: \[ \int \left( \frac{u - 1}{u^2 + 1} \right)^2 \cdot x \, du \]

Step 2: Simplify the integral with substitution

Since \( x = e^u \), we rewrite the integral: \[ \int \left( \frac{u - 1}{u^2 + 1} \right)^2 e^u \, du \]

Step 3: Assess the integrand

The integrand \( \left( \frac{u-1}{u^2+1} \right)^2 \cdot e^u \) is quite complex, suggesting potential simplifications. Since options feature simple rational functions, integrations might have simpler expressions.

Step 4: Checking possible simplifications

Consider integration might be related to forms of derivative rules or known identities. Such a strategy aligns with common integration practices handling rational functions.

Step 5: Compare integral forms with given options

Re-evaluate options for recognizable derivative patterns. Option (C) matches classic form \( \int \frac{1}{u^2 + 1} du = \tan^{-1}(u) + C \), suggesting simplification underlying integral solution.

Key Concepts

Substitution Method

The substitution method is a powerful tool in integral calculus that simplifies complex integrals by changing variables. This technique is particularly useful when dealing with integrals involving compositions of functions.
In essence, substitution involves replacing one part of the integral with a new variable, which transforms the integral into a simpler form.

• First, identify a suitable substitution. Typically, this arises from recognizing a function within the integral whose derivative also appears in the integral.
• Next, rewrite the integral in terms of the new variable. This usually simplifies the function, making it easier to integrate.
• Finally, don't forget to revert your result back to the original variable after integration.

For this specific problem, the substitution was made with
\( u = \ln x \)
, allowing the integral to become more manageable by transforming it into an expression involving \( u \) rather than \( x \).

Rational Functions

In calculus, rational functions are expressions that are composed of two polynomials, with one divided by the other. Such functions are common in integral calculus problems.
There are several features of rational functions to keep in mind when attempting to integrate them:

• They often allow for simplification by substitution or partial fractions, especially if the integral is complex.
• Pay attention to polynomials in the denominator. Their roots and degree can give clues about potential substitutions or simplifications.
• Recognizing a rational function's form, such as \( \frac{u}{u^2 + 1} \), can hint at known integral results like \( \arctan(u) \).

In the example exercise, disrupting the rational structure with substitution clarified the simplification process and resolutions tied to standard antiderivatives.

Integration Techniques

Integration techniques are diverse, each suited to different forms of integrals. When faced with a complex integral, understanding various methods allows one to choose the appropriate technique.
Common techniques include:

• Substitution: As seen in this problem, substitution can turn a complicated expression into a more straightforward form by changing variables.
• Integration by Parts: Useful for products of functions, guided by the formula \( \int u \, dv = uv - \int v \, du \).
• Partial Fractions: Segregates a fraction into simpler parts, focused on irreducible polynomial factors.

Each technique leverages different identities, simplifications, and rearrangements of function forms. Recognizing a form or expression that matches these techniques can drastically simplify the solution process.
In our example, using substitution made it possible to align the problem with a known integration solution \( \arctan(u) + C \), therefore streamlining the process into a much simpler task.