Solving The Indefinite Integral Of (sin 3x + Cos X) / (cos² X + Sin X + 1)
Introduction
In the realm of calculus, integral calculus stands as a fundamental pillar, concerned with the accumulation of quantities and the determination of areas under curves. Among the various facets of integral calculus, the quest for indefinite integrals holds a prominent position. An indefinite integral, often referred to as the antiderivative, represents a function whose derivative is the given function. This article embarks on a journey to unravel the indefinite integral of a specific trigonometric expression, namely, (sin 3x + cos x) / (cos² x + sin x + 1) with respect to x. This exploration will involve employing a combination of trigonometric identities, algebraic manipulations, and strategic substitutions to arrive at a solution. This problem serves as an excellent illustration of the techniques involved in solving complex integrals, which are often encountered in various fields of science and engineering.
The challenge at hand lies in the intricate nature of the integrand, (sin 3x + cos x) / (cos² x + sin x + 1). The presence of trigonometric functions, both in the numerator and the denominator, necessitates a meticulous approach. The path to the solution involves a series of strategic steps, each designed to simplify the expression and make it more amenable to integration. Initially, we will leverage trigonometric identities to rewrite sin 3x in terms of sin x and cos x. This transformation will pave the way for further simplifications and substitutions. Subsequently, we will explore the possibility of employing substitution techniques to reduce the complexity of the integral. By judiciously selecting a suitable substitution, we aim to transform the integral into a more manageable form, one that can be readily evaluated using standard integration formulas. Throughout this process, we will remain vigilant in preserving the accuracy and validity of each step, ensuring that the final result is a true representation of the indefinite integral.
Trigonometric Transformations
To embark on this integration endeavor, our initial focus rests on simplifying the numerator. The sin 3x term presents an opportunity for strategic manipulation. By invoking the trigonometric identity for the sine of a triple angle, we can express sin 3x in terms of sin x and cos x. This identity, a cornerstone of trigonometric transformations, states that sin 3x = 3 sin x - 4 sin³ x. Armed with this identity, we can rewrite the numerator as 3 sin x - 4 sin³ x + cos x. This transformation marks a crucial step forward, as it lays the foundation for subsequent simplifications and substitutions.
Now, let's delve into the denominator, cos² x + sin x + 1. Our goal is to express the denominator in a form that aligns with the transformed numerator, facilitating potential cancellations or substitutions. To achieve this, we can leverage the fundamental trigonometric identity, sin² x + cos² x = 1. By rearranging this identity, we can express cos² x as 1 - sin² x. Substituting this expression into the denominator, we obtain 1 - sin² x + sin x + 1, which can be further simplified to 2 + sin x - sin² x. This manipulation proves to be a strategic move, as it casts the denominator in terms of sin x, mirroring the presence of sin x terms in the transformed numerator.
With both the numerator and the denominator expressed in terms of sin x and cos x, we can now rewrite the original integral as:
∫ (3 sin x - 4 sin³ x + cos x) / (2 + sin x - sin² x) dx
This transformed integral, while still intricate, presents a more structured form. The presence of common trigonometric functions in both the numerator and the denominator hints at the possibility of employing substitution techniques to further simplify the expression. In the upcoming sections, we will explore various substitution strategies, carefully selecting the one that promises the most efficient path towards the solution.
Strategic Substitution
As we delve deeper into the integration process, the strategic application of substitution techniques emerges as a pivotal approach. The transformed integral, ∫ (3 sin x - 4 sin³ x + cos x) / (2 + sin x - sin² x) dx, presents a compelling case for substitution, given the presence of related trigonometric functions in both the numerator and the denominator. Our aim is to identify a substitution that simplifies the integral, transforming it into a more manageable form.
A closer examination of the integral reveals a potential candidate for substitution: u = sin x. This substitution is motivated by the fact that the derivative of sin x is cos x, which appears prominently in the numerator. By choosing this substitution, we aim to eliminate the sin x terms in the denominator and simplify the numerator. To implement this substitution, we must also express dx in terms of du. Differentiating u = sin x with respect to x, we obtain du/dx = cos x, which implies that dx = du / cos x.
With the substitution u = sin x and dx = du / cos x in hand, we can now transform the integral. Substituting u for sin x in the integral, we obtain:
∫ (3u - 4u³ + cos x) / (2 + u - u²) * (du / cos x)
Notice that the cos x terms in the numerator and denominator cancel each other out, further simplifying the integral. This cancellation is a direct consequence of our strategic substitution, highlighting the power of choosing the right substitution to simplify complex integrals. The transformed integral now reads:
∫ (3u - 4u³) / (2 + u - u²) du
This integral, expressed in terms of the variable u, appears significantly simpler than the original integral. The trigonometric functions have been eliminated, leaving us with a rational function to integrate. However, the rational function is still not in a readily integrable form. The denominator, 2 + u - u², is a quadratic expression, which suggests that we may need to employ partial fraction decomposition to further simplify the integral. In the subsequent sections, we will explore the application of partial fraction decomposition to break down the rational function into simpler fractions, paving the way for straightforward integration.
Partial Fraction Decomposition
Having successfully transformed the original integral into ∫ (3u - 4u³) / (2 + u - u²) du, our next challenge lies in integrating the rational function (3u - 4u³) / (2 + u - u²) . The presence of a quadratic expression in the denominator suggests that partial fraction decomposition may be a suitable technique to employ. Partial fraction decomposition is a powerful algebraic technique that allows us to break down complex rational functions into simpler fractions, which are often easier to integrate.
Before we can apply partial fraction decomposition, we need to ensure that the degree of the numerator is less than the degree of the denominator. In our case, the degree of the numerator (3u - 4u³) is 3, while the degree of the denominator (2 + u - u²) is 2. This means we need to perform polynomial long division before we can proceed with partial fraction decomposition. Dividing -4u³ + 3u by -u² + u + 2, we obtain:
-4u³ + 3u = (4u - 4)(-u² + u + 2) + (-5u + 8)
This division allows us to rewrite the integrand as:
(3u - 4u³) / (2 + u - u²) = 4u - 4 + (-5u + 8) / (2 + u - u²)
Now, the rational function (-5u + 8) / (2 + u - u²) has a numerator of degree 1 and a denominator of degree 2, satisfying the condition for partial fraction decomposition. To proceed, we first factor the denominator: 2 + u - u² = -(u - 2)(u + 1). This factorization allows us to express the rational function as a sum of simpler fractions:
(-5u + 8) / (2 + u - u²) = A / (u - 2) + B / (u + 1)
where A and B are constants that we need to determine. To find these constants, we multiply both sides of the equation by the common denominator, (u - 2)(u + 1), yielding:
-5u + 8 = A(u + 1) + B(u - 2)
To solve for A and B, we can use the method of strategic values. Setting u = 2, we eliminate the term involving B and obtain:
-5(2) + 8 = A(2 + 1) => -2 = 3A => A = -2/3
Similarly, setting u = -1, we eliminate the term involving A and obtain:
-5(-1) + 8 = B(-1 - 2) => 13 = -3B => B = -13/3
With the constants A and B determined, we can now express the rational function as a sum of simpler fractions:
(-5u + 8) / (2 + u - u²) = (-2/3) / (u - 2) + (-13/3) / (u + 1)
This decomposition marks a significant step forward, as it transforms the complex rational function into a sum of simpler fractions that are readily integrable. In the next section, we will integrate these simpler fractions, ultimately arriving at the indefinite integral of the original expression.
Integration of Simpler Fractions
With the rational function successfully decomposed into simpler fractions, we are now poised to integrate each fraction individually. The integral we are tackling is:
∫ (3u - 4u³) / (2 + u - u²) du = ∫ [4u - 4 + (-2/3) / (u - 2) + (-13/3) / (u + 1)] du
This integral can be broken down into a sum of simpler integrals:
∫ (3u - 4u³) / (2 + u - u²) du = ∫ (4u - 4) du + ∫ (-2/3) / (u - 2) du + ∫ (-13/3) / (u + 1) du
Each of these integrals can be evaluated using standard integration techniques. The first integral, ∫ (4u - 4) du, is a straightforward application of the power rule for integration:
∫ (4u - 4) du = 2u² - 4u + C₁
where C₁ is the constant of integration.
The second integral, ∫ (-2/3) / (u - 2) du, can be evaluated using a simple substitution. Let v = u - 2, then dv = du, and the integral becomes:
∫ (-2/3) / (u - 2) du = (-2/3) ∫ (1/v) dv = (-2/3) ln|v| + C₂ = (-2/3) ln|u - 2| + C₂
where C₂ is the constant of integration.
Similarly, the third integral, ∫ (-13/3) / (u + 1) du, can be evaluated using a similar substitution. Let w = u + 1, then dw = du, and the integral becomes:
∫ (-13/3) / (u + 1) du = (-13/3) ∫ (1/w) dw = (-13/3) ln|w| + C₃ = (-13/3) ln|u + 1| + C₃
where C₃ is the constant of integration.
Combining these results, we obtain the indefinite integral in terms of u:
∫ (3u - 4u³) / (2 + u - u²) du = 2u² - 4u - (2/3) ln|u - 2| - (13/3) ln|u + 1| + C
where C = C₁ + C₂ + C₃ is the overall constant of integration. However, our original integral was in terms of x, so we need to substitute back u = sin x to express the result in terms of x.
Back-Substitution and Final Result
Having successfully integrated the simpler fractions, we now stand at the precipice of our final goal: expressing the indefinite integral in terms of the original variable, x. Recall that we employed the substitution u = sin x to simplify the integral. To revert to the original variable, we must replace u with sin x in the result we obtained in the previous section.
The indefinite integral in terms of u is:
∫ (3u - 4u³) / (2 + u - u²) du = 2u² - 4u - (2/3) ln|u - 2| - (13/3) ln|u + 1| + C
Substituting u = sin x into this expression, we obtain:
∫ (sin 3x + cos x) / (cos² x + sin x + 1) dx = 2 sin² x - 4 sin x - (2/3) ln|sin x - 2| - (13/3) ln|sin x + 1| + C
This expression represents the indefinite integral of the original function, (sin 3x + cos x) / (cos² x + sin x + 1), with respect to x. The constant of integration, C, signifies the family of functions that have the same derivative as the integrand. The presence of the natural logarithm terms reflects the nature of the partial fraction decomposition and the subsequent integration of the simpler fractions.
This final result encapsulates the culmination of our efforts, a testament to the power of strategic substitutions, trigonometric transformations, and partial fraction decomposition in the realm of integral calculus. The journey from the initial intricate integral to this final expression underscores the importance of meticulous step-by-step problem-solving and the judicious application of mathematical techniques.
Conclusion
In this exploration, we embarked on a quest to determine the indefinite integral of the trigonometric expression (sin 3x + cos x) / (cos² x + sin x + 1) with respect to x. This endeavor served as a compelling illustration of the techniques involved in tackling complex integrals, a common challenge in various fields of science and engineering. The journey involved a strategic blend of trigonometric transformations, algebraic manipulations, and judicious substitutions, each step carefully executed to simplify the integral and make it more amenable to evaluation.
We initiated our approach by leveraging trigonometric identities to rewrite sin 3x in terms of sin x and cos x, a crucial step that paved the way for subsequent simplifications. We then strategically employed the substitution u = sin x, a choice motivated by the presence of cos x in the numerator and the desire to eliminate the sin x terms in the denominator. This substitution proved to be a pivotal move, transforming the integral into a rational function in terms of u.
However, the rational function was not yet in a readily integrable form. To overcome this hurdle, we turned to the powerful technique of partial fraction decomposition. This technique allowed us to break down the complex rational function into simpler fractions, each of which could be integrated using standard integration formulas. The process of partial fraction decomposition involved careful algebraic manipulation and the determination of constants, a testament to the importance of precision in mathematical problem-solving.
Finally, we integrated the simpler fractions, obtaining the indefinite integral in terms of u. To complete the solution, we back-substituted u = sin x, expressing the indefinite integral in terms of the original variable, x. The final result, a complex expression involving trigonometric functions and natural logarithms, stands as a testament to the power of the techniques employed and the intricacies of integral calculus.
This exploration underscores the importance of a systematic and strategic approach to solving complex integrals. The ability to identify suitable substitutions, leverage trigonometric identities, and apply partial fraction decomposition are essential skills for anyone venturing into the realm of integral calculus. The journey from the initial intricate integral to the final solution serves as a valuable learning experience, reinforcing the fundamental principles of calculus and highlighting the beauty and elegance of mathematical problem-solving.
