Problem 12 - Calculus 2 Course in Mathematics for the IIT-JEE and Other Engineering Exams

Question

. Intcgrals of the Form \(\int \frac{a \cos x+b \sin x}{c \cos x+d \sin x} d x\) Working Rule (i) Put Numerator: \(\lambda\) (denominator) \(+\mu\) (derivative of denominator) \(a \cos x+b \sin x=\lambda(c \cos x+d \sin x)+\) \(\mu(-c \sin x+d \cos x)\) (ii) Fquate cocfficients of \(\sin x\) and \(\cos x\) on both sides and find the values of \(\lambda\) and \(\mu\). (iii) Split the given integral into two integrals and cvaluate cach integral separately, i.c. \(\int \frac{a \cos x+b \sin x}{c \cos x+d \sin x} d x=\lambda\) \(\int 1 d x+\mu \int \frac{-c \sin x+d \cos x}{c \cos x+d \sin x} d x\) \(=\lambda x+\mu \log |a \cos x+b \sin x| .\) (iv) Substitute the values of \(\lambda\) and \(\mu\) found in step 2 .

Step-by-Step Solution

Answer

Find \(\lambda\) and \(\mu\) from coefficients, split the integral, solve each part, and combine results.

Step 1: Setup the Numerator Equation

We start by expressing the numerator as a linear combination of the functions in the denominator and its derivative. Let the numerator, \(a \cos x + b \sin x\), equal \(\lambda (c \cos x + d \sin x) + \mu (-c \sin x + d \cos x)\). This expression lets us apply the integration rule provided in the problem.

Step 2: Equate Coefficients

Expand both sides of the equation from Step 1. On the left side, we have \(a \cos x + b \sin x\). On the right side, after distribution, it results in \((\lambda c + \mu d) \cos x + (\lambda d - \mu c) \sin x\). By equating coefficients of \(\cos x\) and \(\sin x\), we get two equations: \(a = \lambda c + \mu d\) and \(b = \lambda d - \mu c\).

Step 3: Solve for \(\lambda\) and \(\mu\)

Solve the system of linear equations obtained in Step 2 for \(\lambda\) and \(\mu\). Use methods such as substitution or elimination to find explicit values for these variables, ensuring that they satisfy both equations simultaneously.

Step 4: Split the Integral

Use the values of \(\lambda\) and \(\mu\) to decompose the original integral as specified: \(\int \frac{a \cos x + b \sin x}{c \cos x + d \sin x} dx = \lambda \int 1 \, dx + \mu \int \frac{-c \sin x + d \cos x}{c \cos x + d \sin x} dx\).

Step 5: Evaluate Each Integral

The first integral is straightforward: \(\lambda \int 1 \, dx = \lambda x\). The second integral evaluates to \(\mu \log |c \cos x + d \sin x|\), using the derivative of the denominator in this form. Combine these to get the overall integral result.

Step 6: Final Substitution

Substitute the values obtained for \(\lambda\) and \(\mu\) into the solution: \(\lambda x + \mu \log |c \cos x + d \sin x| + C\), where \(C\) is the constant of integration.

Key Concepts

Trigonometric Integration

Integrating expressions that involve trigonometric functions can often appear more daunting than they really are. In particular, when faced with integrals of the form \( \int \frac{a \cos x + b \sin x}{c \cos x + d \sin x} \, dx \), the trick is to utilize a strategic approach. The key is to express the numerator as a linear combination of the trigonometric terms in the denominator and its derivative.

To find this expression, rewrite the numerator \( a \cos x + b \sin x \) as \( \lambda(c \cos x + d \sin x) + \mu(-c \sin x + d \cos x) \). This step is crucial and allows you to separate the integral into more manageable parts. Once in this form, solving the integral becomes a matter of basic integration techniques.

By turning complex trigonometric expressions into easier components with known integrative solutions, we simplify the problem. These trigonometric techniques open a realm where challenging integrals become easier to solve by inspection or simple calculations.

Linear Combination

The concept of a linear combination is a cornerstone in solving integrals like the one presented. Here, the numerator \( a \cos x + b \sin x \) is expressed as a combination of basis functions from the denominator \( c \cos x + d \sin x \) and its derivative \(-c \sin x + d \cos x \).

This transformation allows us to manipulate the equation based on the coefficients of \( \cos x \) and \( \sin x \). When we equate these coefficients on both sides, we derive a system of linear equations:

• \( a = \lambda c + \mu d \)
• \( b = \lambda d - \mu c \)

These equations are the key to finding the values of \( \lambda \) and \( \mu \), unlocking the pathway to split the original integral into simpler parts.

The use of linear combination here is not only a mathematical tool but a method to simplify and strategize the resolution of complex integral problems. It's these methods that help reduce computational errors and foster a deeper understanding of algebraic relationships.

Solving Systems of Equations

When faced with a system of linear equations from the process of equating coefficients, solving for unknowns like \( \lambda \) and \( \mu \) becomes necessary. This is where solving systems of equations comes into play. You can use various methods, such as substitution or elimination, to find these values.

For the equations \( a = \lambda c + \mu d \) and \( b = \lambda d - \mu c \), solving them gives you the coefficients needed to decompose the integral. It’s crucial that the solution satisfies both equations simultaneously, ensuring consistency and correctness.

By solving these equations efficiently, you can precisely backtrack and use these coefficients for integration. This is not just about finding values but understanding how these values interplay to influence the integral's solution. Mastering systems of equations aids in executing this understanding and solving similar problems with ease.