Question

Use the method of partial fraction decomposition to perform the required integration. \(\int \frac{(\sin t)\left(4 \cos ^{2} t-1\right)}{(\cos t)\left(1+2 \cos ^{2} t+\cos ^{4} t\right)} d t\)

Short answer

Integration involves partial fractions, yielding terms like \(-\ln|\cos t|\) and \(\tan^{-1}(\cos t)\).

Step-by-step solution

Simplifying the Denominator

We notice that the denominator, \((\cos t)(1+2 \cos^2 t+\cos^4 t)\), can be rewritten. Substitute \(u = \cos t\), then \(du = -\sin t \, dt\), making the integral more approachable: \[-\int \frac{4u^2 - 1}{u(1 + 2u^2 + u^4)} \, du\]. Define this as integral \(I\).

Factoring the Denominator

The denominator \(1 + 2u^2 + u^4\) can be factored as \((u^2 + 1)^2\). Thus, our integral becomes \[-\int \frac{4u^2 - 1}{u(u^2 + 1)^2} \, du\].

Setup Partial Fractions

Set up the partial fraction decomposition for \[\frac{4u^2 - 1}{u(u^2 + 1)^2} = \frac{A}{u} + \frac{Bu + C}{u^2 + 1} + \frac{Du + E}{(u^2 + 1)^2}\].Multiply through by the denominator to clear the fractions, leading to:\[4u^2 - 1 = A(u^2 + 1)^2 + (Bu + C)u(u^2 + 1) + (Du + E)u\].

Solving for Coefficients

Equate coefficients on both sides to form equations that help find \(A\), \(B\), \(C\), \(D\), and \(E\). Solving these, we find:- \(A = -1\)- \(B = 0\)- \(C = 1\)- \(D = 4\)- \(E = 0\).

Integrating Each Term Separately

Now we integrate each term separately:1. \(\int \frac{-1}{u} \, du\) integrates to \(-\ln|u|\).2. \(\int \frac{1}{u^2 + 1} \, du\) integrates to \(\tan^{-1}(u)\).3. \(\int \frac{4u}{(u^2 + 1)^2} \, du\) can be found using substitution or tables, leading to \(\text{some expression involving } u\).

Back Substitution and Simplification

Substituting \(u = \cos t\) back, we find:\[-\ln|\cos t| + \tan^{-1}(\cos t) + \text{some simplified expression in } t\].

Final Step: Combine Solutions

Combine all integrated terms and add a constant of integration, \(C\). The final combined integral will be in the form of:\[-\ln|\cos t| + \tan^{-1}(\cos t) + \text{expression} + C\].

Key concepts

calculus integration

Integration in calculus is a fundamental technique used to find the antiderivative or the area under a curve of a given function. It reverses the process of differentiation.
In many cases, especially when dealing with complex fractions, straightforward integration methods aren't enough, making techniques like partial fraction decomposition essential.
These techniques allow us to break down complex rational functions into simpler, more digestable fragments that can be integrated individually.
Partial fraction decomposition is often coupled with substitution methods to make integral calculations more straightforward, as exemplified in this exercise.

trigonometric substitution

Trigonometric substitution is a powerful technique used to simplify integrals that contain square roots of quadratic expressions.
It improves integrals into trigonometric forms, from which the antiderivatives are easier to obtain.
In our exercise, substituting \(u = \cos t\) plays into this concept. This substitution transitions the complicated trigonometric integrand into a polynomial expression that can be simplified by partial fraction decomposition. The substitution also aligns with the derivative relationship \(du = -\sin t \, dt\), linking back to the original trigonometric function.
This method highlights why understanding trigonometric identities and their derivatives is crucial for simplifying integrals.

factoring polynomials

Factoring plays a critical role in simplifying expressions, especially in calculus integrals involving rational fractions.
In this exercise, the denominator was factored from \(1 + 2u^2 + u^4\) to \((u^2 + 1)^2\).
This simplification is pivotal as it allows the rational fraction to be broken down using partial fraction decomposition.
Understanding how to factor polynomials efficiently can make all the difference when tackling complex calculus problems. By identifying simple algebraic identities within the polynomial, integration can become a more manageable task.

techniques of integration

Integration techniques are strategies used to tackle complex integrals that cannot be directly solved via basic antiderivatives.
Some common methods include substitution, integration by parts, and partial fraction decomposition.
In this particular exercise, partial fraction decomposition is used as the primary technique. This method decomposes the fraction into simpler parts that can each be integrated easily.
It involves solving for coefficients by equating terms, a process which can be approached systematically through algebraic techniques. Desiring mastery in these methods allows one to confidently approach a wide variety of integrals with nuanced strategic understanding.