Problem 102 - A Complete Resource Book in Mathematics for JEE Main 2017

Question

If the integral \(\int \frac{5 \tan x}{\tan x-2} d x=x+a \ln |\sin x-2 \cos x|+k\) then a is equal to (A) \(-1\) (B) \(-2\) (C) 1 (D) 2

Step-by-Step Solution

Answer

The coefficient \( a \) is \(-2\), so the answer is (B) \(-2\).

Step 1: Analyzing the Given Integral

The integral provided is \( \int \frac{5 \tan x}{\tan x-2} dx \). We need to find the value of \( a \) such that the integral results in the given expression \( x + a \ln |\sin x - 2 \cos x| + k \).

Step 2: Using Substitution for Simplification

Notice that the expression inside the logarithm is \( \sin x - 2 \cos x \). Consider using substitution techniques to simplify the integral. Let \( u = \sin x - 2 \cos x \). Then, find \( du = (\cos x + 2 \sin x) dx \).

Step 3: Differentiating and Rearranging Terms

From \( \tan x = \frac{\sin x}{\cos x} \), we can rewrite the integral to fit the substitution. Also, observe that to substitute \( \tan x - 2 \), we use the identity \( 1 + \tan^2 x = \sec^2 x \) to differentiate. Make appropriate substitutions to simplify the expression.

Step 4: Integrating and Comparing

Integrate the new expression obtained after substitution. Compare the form of the integrated function with \( x + a \ln |\sin x - 2 \cos x| + k \). Especially, observe the coefficient in front of the logarithmic term which corresponds to \( a \).

Step 5: Identifying Coefficient 'a'

After integration and comparison, the coefficient of the \( \ln |\sin x - 2 \cos x| \) term is \(-2\). Thus, after all simplifications, \( a = -2 \).

Key Concepts

Integrals

The concept of integrals is central in calculus, often used to find areas under curves, among other applications. An integral can be thought of as the opposite of a derivative. It helps in determining the accumulation of quantities, such as areas, totals, and sums. There are two main types of integrals you will encounter:

• Definite Integrals: These have upper and lower limits, and are used to calculate the exact area under a curve between two points.


• Indefinite Integrals: These do not have limits, and they represent a family of functions or the antiderivatives.

In your studies, you might frequently find integrals expressed in the form \( \int{f(x) dx} \). Solving these often requires a range of techniques, such as substitution or utilizing trigonometric identities, especially when dealing with complex expressions.

Substitution Method

The substitution method is a powerful tactic in calculus that simplifies the integration process. It involves changing variables to transform a difficult integral into a more manageable one. Here's how you generally approach this method:

• Select a Substitution: Choose a new variable \( u \) to replace a part of the integral, often a complicated expression or one within a function.


• Differentiate: Calculate the derivative \( du \) of your substitution with respect to \( x \). This helps in replacing \( dx \) eventually.


• Rewrite the Integral: Substitute the terms of the integral with expressions in terms of \( u \) and \( du \).


• Integrate and Back-Substitute: Perform the integration in terms of \( u \), then revert back to the original variable \( x \) using the substitution equation.

An example from the problem above is using \( u = \sin x - 2 \cos x \) to ease the integration process. Correct substitutions simplify the problem greatly, making it easier to identify components like the coefficient in this case.

Trigonometric Identities

Trigonometric identities are equations that are true for all angles and are invaluable tools in simplifying integrals involving trigonometric functions. They assist in rewriting complex expressions into simpler forms, which can be readily integrated.

• Pythagorean Identity: One of the most useful identities is \(1 + \tan^2 x = \sec^2 x \). It helps in expressing \( \tan^2 x \) in terms of \( \sec^2 x \).


• Sum-to-Product Formulas: These identities transform sums of trigonometric functions into products, often simplifying the integrals considerably.


• Basic Synonyms: Understand relationships like \( \tan x = \frac{\sin x}{\cos x} \) to replace terms effectively.

In the given exercise, utilizing the identity \(1 + \tan^2 x = \sec^2 x \) was key in making substitutions that eventually led to identifying the coefficient \(a\). Mastering these identities arms you with tools to handle even the most complex integrals effectively.