Problem 75 - A Complete Resource Book in Mathematics for JEE Main

Question

If \(\int \frac{x \log \left(x+\sqrt{1+x^{2}}\right)}{\sqrt{1+x^{2}}} d x\) \(=A \sqrt{1+x^{2}} \log \left(x+\sqrt{1+x^{2}}\right)+B x+C\), then (A) \(A=-1\) (B) \(B=-1\) (C) \(A=1\) (D) none of these

Step-by-Step Solution

Answer

A = -1, answer is (A).

Step 1: Identify the Integral Components

Consider the integral \( \int \frac{x \log \left(x+\sqrt{1+x^{2}}\right)}{\sqrt{1+x^{2}}} \, dx \). We need to identify components of the function to use integration techniques effectively. Notice that \( x+\sqrt{1+x^2} \) is a common substitution in integrals involving \( \sqrt{1+x^2} \).

Step 2: Use Substitution

Let \( u = x + \sqrt{1+x^2} \). Then, derivative \( du = \left(1 + \frac{x}{\sqrt{1+x^2}}\right)dx = \frac{x + \sqrt{1+x^2}}{\sqrt{1+x^2}} dx = \frac{u}{\sqrt{1+x^2}} dx \). Simplifying, \( x = \frac{u^2 - 1}{2u} \).

Step 3: Express the Integral in Terms of u

Substitute the expression of \( x \) and \( dx \) in terms of \( u \) into the integral: \( \int \frac{(\frac{u^2 - 1}{2u}) \log(u)}{\sqrt{1+x^2}} \frac{\sqrt{1+x^2}}{u} du \). This simplifies to \( \frac{1}{2} \int (u - \frac{1}{u}) \log(u) \, du \).

Step 4: Break Down the Integral

The integral becomes \( \frac{1}{2} \int u \log(u) \, du - \frac{1}{2} \int \frac{\log(u)}{u} \, du \). Separate the integrals: \( I_1 = \frac{1}{2} \int u \log(u) \, du \) and \( I_2 = -\frac{1}{2} \int \frac{\log(u)}{u} \, du \).

Step 5: Solve Each Integral Separately

Using integration by parts for \( I_1 \): let \( v = \log(u) \) and \( dw = u \, du \); \( dv = \frac{1}{u} \, du \) and \( w = \frac{u^2}{2} \). Therefore, \( I_1 = \frac{1}{2} \left( \frac{u^2}{2} \log(u) - \int \frac{u^2}{2} \frac{1}{u} \, du \right) = \frac{1}{2} \left( \frac{u^2}{2} \log(u) - \frac{u^2}{4} \right) \). For \( I_2 \), the integral is simple resulting in \( -\frac{1}{4}u^2 + C \).

Step 6: Combine the Results and Compare with Given Expression

Combine the results of the integrals: \( A \sqrt{1+x^2} \log(x+\sqrt{1+x^2}) + Bx + C \). Compare: \( \frac{u^2}{4} \log(u) - \frac{u^2}{4} - \frac{1}{4} \frac{u^2}{2} \) with \( A \sqrt{1+x^2} \log (x+\sqrt{1+x^2}) + Bx + C \). From here, determine the values of \( A \) and \( B \).

Step 7: Conclusion from Comparison

Comparing both sides, you find that \( A = -1 \) by matching coefficients with \( \frac{u^2}{4} \log(u) - \frac{u^2}{4} \), as the leading term \( A \) in the given expression affects the coefficient of the logarithmic term.

Key Concepts

Integration Techniques

Integral calculus often requires different techniques to solve complex integrals. One effective approach is to identify the structure and patterns within the given integral. In the exercise, the integral \( \int \frac{x \log \left(x+\sqrt{1+x^{2}}\right)}{\sqrt{1+x^{2}}} \, dx \) involves recognizing the common pattern \( x+\sqrt{1+x^2} \), which helps in choosing an appropriate substitution. This pattern hints at a suitable variable substitution, which simplifies the integration process greatly.

Here are some common integration techniques:

• Substitution: Replace a complex part of the integral with a single variable to simplify integration.
• Partial fractions: Decompose rational expressions for easier integration.
• Integration by parts: Useful for integrals involving a product of functions, derived from the product rule of differentiation.

Recognizing which technique to apply is often the key to solving an integral successfully.

Logarithmic Integration

Logarithmic integration manages integrals that involve logarithmic functions, like the exercise integral which incorporates \( \log \left(x+\sqrt{1+x^{2}}\right) \). The logarithm can often complicate integration, but by using substitution and recognizing patterns, the integration can become more straightforward.

In this specific case, once substitution turns the integral into terms involving \( u \), the problem simplifies to integrating functions like \( u \log(u) \) and \( \frac{\log(u)}{u} \).

Key points in logarithmic integration:

• Simplify: Aim to express the logarithmic term in the simplest possible form.
• Use substitution effectively: Transform logarithmic functions and their differentials to reduce complexity.
• Separation of integrals: Break down the integral into simpler components that are easier to handle.

This method can transform otherwise daunting logarithmic integrals into a series of manageable steps.

Integration by Parts

Integration by parts is a technique derived from the product rule for differentiation. It's particularly useful for integrating products of functions, such as \( u \log(u) \) in this exercise.

The basic formula for integration by parts is: \[ \int u \, dv = uv - \int v \, du \]where you choose \( u \) and \( dv \) such that differentiating \( u \) and integrating \( dv \) simplifies the integral.

In our step-by-step solution, we've set \( v = \log(u) \) and \( dw = u \, du \). This allows us to rewrite the integral in more manageable terms, leading to the solution more efficiently.

Tips for using integration by parts include:

• Choose \( u \) and \( dv \) wisely: Ideally, \( u \) is something that simplifies upon differentiation, and \( dv \) is easily integrable.
• Don't hesitate to repeat: Sometimes integration by parts needs to be applied more than once to solve an integral.
• Balance complexity: Aim to reduce the complexity of the integral with each step.

Mastering this technique provides a powerful tool for tackling a wide variety of challenging integrals.