Problem 99 - A Complete Resource Book in Mathematics for JEE Main

Question

\(\int \frac{d x}{\cos x+\sqrt{3} \sin x}\) equals (A) \(\frac{1}{2} \log \tan \left(\frac{x}{2}+\frac{\pi}{12}\right)+\mathrm{c}\) (B) \(\frac{1}{2} \log \tan \left(\frac{x}{2}-\frac{\pi}{12}\right)+\mathrm{c}\) (C) \(\operatorname{logtan}\left(\frac{-}{2}-\frac{\underline{\phantom{xx}}}{12}\right)+\mathrm{c}\) (D) \(\log \tan \left(\frac{x}{2}-\frac{\pi}{12}\right)+\mathrm{c}\)

Step-by-Step Solution

Answer

The correct answer is (B).

Step 1: Rewrite the Trigonometric Expression

The integral is given by \( \int \frac{d x}{\cos x + \sqrt{3} \sin x} \). First, observe the denominator. Notice that the expression can be expressed as a combination of sine and cosine: \( a \cos x + b \sin x \), where \( a = 1 \) and \( b = \sqrt{3} \).

Step 2: Use Trigonometric Identities to Simplify

To simplify \( a \cos x + b \sin x \), we can express it in the form \( R \cos(x - \alpha) \), where \( R = \sqrt{a^2 + b^2} = \sqrt{1^2 + (\sqrt{3})^2} = 2 \). The angle \( \alpha \) is given by \( \tan \alpha = \frac{b}{a} = \frac{\sqrt{3}}{1} = \sqrt{3} \), thus \( \alpha = \frac{\pi}{3} \).

Step 3: Substitute and Simplify the Integral

The integral becomes \( \int \frac{d x}{2 \cos(x - \frac{\pi}{3})} \). Simplifying, we get \( \frac{1}{2} \int \sec(x - \frac{\pi}{3}) \, d x \).

Step 4: Solve the Integral

The integral \( \int \sec(x - \frac{\pi}{3}) \, dx \) can be solved using the known integral of secant: \( \int \sec u \, du = \ln | \tan(\frac{u}{2} + \frac{\pi}{4}) | + C \). Therefore, \( \int \sec(x - \frac{\pi}{3}) \, dx = \ln | \tan(\frac{x - \frac{\pi}{3}}{2} + \frac{\pi}{4}) | + C \).

Step 5: Simplify Final Expression

Simplifying the expression further, it can be written as \( \ln | \tan(\frac{x}{2} - \frac{\pi}{6}) | + C \), due to the trigonometric transformations.Taking the factor \( \frac{1}{2} \) from Step 3 into account, the final solution is \( \frac{1}{2} \ln | \tan(\frac{x}{2} - \frac{\pi}{12}) | + C \).

Step 6: Match with Provided Options

The solution \( \frac{1}{2} \ln | \tan(\frac{x}{2} - \frac{\pi}{12}) | + C \) matches the option given as (B).

Key Concepts

Trigonometric Identities

When dealing with integrals that involve trigonometric functions, such as in the integral \( \int \frac{d x}{\cos x + \sqrt{3} \sin x} \), leveraging trigonometric identities is vital. These identities can help simplify complex expressions and make integration more manageable.

In this exercise, we notice that the denominator \( \cos x + \sqrt{3} \sin x \) can be represented in a more convenient form using the identity \( R \cos(x - \alpha) \), where \( R \) and \( \alpha \) are constants that depend on the values of the coefficients of \( \cos x \) and \( \sin x \).

This is done by recognizing that any expression \( a \cos x + b \sin x \) can be rewritten as \( R \cos(x - \alpha) \), with \( R = \sqrt{a^2 + b^2} \). In our case, \( a = 1 \) and \( b = \sqrt{3} \), giving \( R = 2 \).

Additionally, \( \tan \alpha = \frac{b}{a} \) helps in finding the angle \( \alpha \). Here, \( \tan \alpha = \sqrt{3} \), so \( \alpha = \frac{\pi}{3} \).

Using these identities simplifies the original integral and sets the stage for applying further integration techniques.

Integration Techniques

To evaluate integrals involving trigonometric functions, it's important to understand certain integration techniques. For this integral, \( \int \frac{d x}{2 \cos(x - \frac{\pi}{3})} \), the substitution method, and trigonometric transformations are key.

After employing trigonometric identities to simplify the expression, the integral turns into \( \frac{1}{2} \int \sec(x - \frac{\pi}{3}) \, d x \).

This expression can be tackled using a known integration formula: \( \int \sec u \, du = \ln | \tan(\frac{u}{2} + \frac{\pi}{4}) | + C \).

Here, substituting \( u = x - \frac{\pi}{3} \), we can directly apply the formula.

It's vital to be familiar with these standard integrals since they provide a toolkit to solve more complex problems. Knowing how and when to apply these techniques can greatly simplify the task at hand, making it approachable and understandable.

Definite Integrals

The integral presented is an indefinite integral, not bounded by specific limits, thus represented with a constant \( C \). However, it's beneficial to briefly understand definite integrals, as they are a logical next step.

Definite integrals calculate the area under a curve within certain bounds, symbolized by putting upper and lower limits. If our expression was bounded in a definite format, the calculation would give a numeric result, unlike the indefinite case, which provides a family of functions.

In practical terms:

• Definite integrals use the same integration techniques but include evaluating the antiderivative at specific points to find the total change or net area under a curve.
• Evaluating definite integrals often involves an "Integration by Parts" or "Substitution" method.
• The result is precise, with no constant of integration \( C \), as it's encapsulated within the bounds.

Understanding both indefinite and definite integrals empowers students to tackle a wide range of problems in calculus, aiding in finding various physical and geometric quantities.