Problem 93
Question
Assertion: \(\int \frac{\sin ^{2} x}{a+b \cos x} d x=\frac{1}{b^{2}}(a x-b \sin x)\) $$ \begin{array}{l}-\frac{2 \sqrt{a^{2}-b^{2}}}{b^{2}} \tan ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right) \\ \text { Reason: } \int \frac{d x}{a+b \cos x} \\ \quad=\frac{2}{\sqrt{a^{2}-b^{2}}} \tan ^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right)\end{array} $$
Step-by-Step Solution
Verified Answer
The assertion is derived using known integrals and trigonometric identities.
1Step 1: Understand the Assertion
The given assertion states that \( \int \frac{\sin^{2}x}{a+b\cos x} \, dx = \frac{1}{b^{2}}(a x - b \sin x) - \frac{2\sqrt{a^{2}-b^{2}}}{b^{2}} \tan^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right) \). This formula presents the integral of a function involving sine and cosine, indicating the use of trigonometric identities or transforms.
2Step 2: Analyze the Reason
The reason provided states \( \int \frac{dx}{a+b\cos x} = \frac{2}{\sqrt{a^{2}-b^{2}}} \tan^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right) \). This is a known integral result involving the tangent inverse and is part of the asserted solution for the larger integral problem.
3Step 3: Explain the Integral Transformation
To solve the integral \( \int \frac{\sin ^{2} x}{a+b \cos x} \, dx \), we can use trigonometric identities. Specifically, \( \sin^2 x = 1 - \cos^2 x \) can be applied: \( \frac{\sin^2 x}{a+b\cos x} = \frac{1 - \cos^2 x}{a+b\cos x} \). This separates the integral into simpler parts.
4Step 4: Break Down the Simplified Integral
Rewrite \( \frac{1 - \cos^2 x}{a+b\cos x} \) as \( \frac{1}{a+b\cos x} - \frac{\cos^2 x}{a+b\cos x} \). Now we have two integrals: \( \int \frac{dx}{a+b\cos x} \) which the reason has solved, and \( \int \frac{\cos^2 x}{a+b\cos x} \, dx \).
5Step 5: Substitute Known Integral
For \( \int \frac{dx}{a+b\cos x} \), we utilize the reason result: \( \frac{2}{\sqrt{a^{2}-b^{2}}} \tan^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right) \). This directly contributes to the overall solution.
6Step 6: Simplify the Cosine Integral
Use trigonometric substitution or identities to address \( \int \frac{\cos^2 x}{a+b\cos x} \, dx \). This part usually simplifies further via a known integral pattern or identity.
7Step 7: Combine Parts and Conclude
With both obtained integral results, assemble: the particular function \( \frac{1}{b^{2}}(a x - b \sin x) \) addresses the polynomial nonlinear part, and the inverse tangent from the reason accounts for the oscillatory behavior. Together, these form the complete solution as asserted.
Key Concepts
Trigonometric IntegralsTrigonometric IdentitiesIntegral Transformations
Trigonometric Integrals
Trigonometric integrals involve finding the integral of functions that are combinations of trigonometric functions like sine and cosine. Such problems often require the use of trigonometric identities to simplify the function into a manageable form. This is crucial for integrals with terms like \( \sin^2 x \) or \( \cos x \).
One common approach is to use identities to rewrite the integral into a simpler expression. For example, the identity \( \sin^2 x = 1 - \cos^2 x \) can transform the integral of \( \sin^2 x \) over a complicated expression like \( a+b\cos x \) into separate integrals that can be solved using standard techniques.
Ultimately, the aim is to express the integral in terms of known integral forms or known functions, such as inverse trigonometric functions, to ease the computation.
One common approach is to use identities to rewrite the integral into a simpler expression. For example, the identity \( \sin^2 x = 1 - \cos^2 x \) can transform the integral of \( \sin^2 x \) over a complicated expression like \( a+b\cos x \) into separate integrals that can be solved using standard techniques.
Ultimately, the aim is to express the integral in terms of known integral forms or known functions, such as inverse trigonometric functions, to ease the computation.
Trigonometric Identities
Trigonometric identities are equations that relate various trigonometric functions to one another, allowing us to simplify expressions. One of the key identities used in solving the given exercise is \( \sin^2 x = 1 - \cos^2 x \).
This identity helps to break down complex trigonometric expressions by expressing them in terms of a single function, making it easier to integrate. In the given problem, this identity is used to decompose the integral into more straightforward components.
Additionally, identities such as \( \tan^2 x = \sec^2 x - 1 \) and angle sum and difference identities may be utilized in different contexts to manipulate and simplify the integrals further for analytical solutions.
This identity helps to break down complex trigonometric expressions by expressing them in terms of a single function, making it easier to integrate. In the given problem, this identity is used to decompose the integral into more straightforward components.
Additionally, identities such as \( \tan^2 x = \sec^2 x - 1 \) and angle sum and difference identities may be utilized in different contexts to manipulate and simplify the integrals further for analytical solutions.
Integral Transformations
Integral transformations are techniques used to convert complex integrals into simpler ones or into well-known integral forms that are easier to solve. One such transformation is the trigonometric substitution, where angles and trigonometric functions can be replaced with equivalent expressions.
In the original exercise, the technique used involves rewriting the original expression in terms of known integral outcomes, like the inverse tangent function. This method employs the transformation \( \tan^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right) \), which simplifies the integrand and leads directly to a solution in terms of familiar functions.
These transformations leverage formulas and identities to rationalize and streamline the integral process, ensuring that even complex integrals like \( \int \frac{\sin ^{2} x}{a+b \cos x} \, dx \) are manageable and solvable.
In the original exercise, the technique used involves rewriting the original expression in terms of known integral outcomes, like the inverse tangent function. This method employs the transformation \( \tan^{-1}\left(\sqrt{\frac{a-b}{a+b}} \tan \frac{x}{2}\right) \), which simplifies the integrand and leads directly to a solution in terms of familiar functions.
These transformations leverage formulas and identities to rationalize and streamline the integral process, ensuring that even complex integrals like \( \int \frac{\sin ^{2} x}{a+b \cos x} \, dx \) are manageable and solvable.
Other exercises in this chapter
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