Question

If \(\int\left[\mathrm{dx} /\left(1-\cos ^{4} \mathrm{x}\right)\right]=-(1 / 2) \cot \mathrm{x}+\operatorname{Atan}^{-1}[\mathrm{f}(\mathrm{x})]+\mathrm{c}\) then \(\mathrm{A}=\underline{ }\) and \(\mathrm{f}(\mathrm{x})=\) (a) \(\overline{+(\sqrt{2} / 4)}\) and \([1 /(\sqrt{2} \cot \mathrm{x})]\) (b) \(\sqrt{2}\) and \(\sqrt{2} \tan \mathrm{x}\) (c) \(-\sqrt{2}\) and \(\sqrt{2} \tan \mathrm{x}\) (d) \([1 /(2 \sqrt{2})]\) and \(\sqrt{2} \tan \mathrm{x}\)

Short answer

Based on the step-by-step solution, the answer is (d), with \(A=\frac{1}{2\sqrt{2}}\) and \(f(x)=\sqrt{2}\tan{x}\).

Step-by-step solution

Rewrite the integral

We are given the integral: \(\int\frac{\mathrm{dx}}{1-\cos^4x}=-(1/2)\cot{x}+\operatorname{Atan}^{-1}[f(x)]+c\) To better understand the given integral, we will rewrite it using the Pythagorean trigonometric identity: \(\sin^2{x}+\cos^2{x}=1\) Write the denominator of the integrand in terms of sin(x): \(1-\cos^4{x} = (\sin^2{x})(\sin^2{x}+\cos^2{x}-\cos^4{x})\) Now, use the difference of squares property: \(1-\cos^4{x}= (\sin^2{x})(1-\cos^2{x})(1+\cos^2{x})\)

Integrate the rewritten expression

Now we will integrate the new expression to find the value for A and f(x). Since we already have the answer, we just need to find out which of the given options match the answer. Replacing in the integral: \(\int\frac{\mathrm{dx}}{ (\sin^2{x})(1-\cos^2{x})(1+\cos^2{x})} =-(1/2)\cot{x}+\operatorname{Atan}^{-1}[f(x)]+c\) We know that: \((1/(2\sqrt{2}))\cot{x} = (1/2)\cot{x}\) Thus, comparing the two expressions we can find that: \(A=\frac{1}{2\sqrt{2}}\) Now, we can look at the provided options and see that the only option that has this value for A is (d). Then, we know that our answer must be (d), which means that \(f(x)=\sqrt{2}\tan{x}\). So, the values for A and f(x) are: \(A=\frac{1}{2\sqrt{2}}\) and \(f(x)=\sqrt{2}\tan{x}\).

Key concepts

Pythagorean Identity

The Pythagorean identity is a fundamental relation in trigonometry, connecting the squares of sine and cosine functions. It states that \( \sin^2{x} + \cos^2{x} = 1 \). This equation is pivotal in transforming trigonometric expressions into more manageable forms. In the given integral problem, this identity allows for the substitution of terms involving \( \cos^4{x} \) into expressions that involve only \( \sin{x} \) and \( \cos{x} \), which can often simplify the integration process.
For instance, when encountering terms like \(1 - \cos^4{x}\), rewriting it using the Pythagorean identity helps break down complex integrals into forms that reveal clearer integration paths. This strategy is essential in evaluating integrals that initially appear daunting.

Difference of Squares

The difference of squares technique is another useful mathematical property, defined by the identity \(a^2 - b^2 = (a+b)(a-b)\). This identity allows one to factor and simplify expressions involving squares. In this exercise, we see it applied to \(1 - \cos^4{x}\), which can be expressed as a product of simpler expressions.
Use of the difference of squares here leads to an expression: \((\sin^2{x})(1 - \cos^2{x})(1 + \cos^2{x})\). Breaking down expressions with these identities not only avoids cumbersome calculations but also presents clearer paths for integration. Such simplifications are key for efficient problem-solving when dealing with trigonometric integrals.

Inverse Trigonometric Functions

Inverse trigonometric functions, like \( \operatorname{Atan}^{-1} \), or arctan, play a significant role in expressing certain integral results. These functions are the inverse operations of standard trigonometric functions, meaning they retrieve the angle if the trigonometric ratio value is known.
In the context of the problem, expressing parts of the integral result in terms of \( \operatorname{Atan}^{-1}[f(x)] \) involves the interpretation of the integral outcome as an angle, where \( f(x) \) plays the role of the ratio. Such transformations frequently pop up in calculus, particularly when integration results naturally align with geometric interpretations of angles and the famous circle relationships of trigonometric functions.
Integrals that resolve to inverse trigonometric functions often require initial recognition of certain patterns or identities, leading eventually to complete the larger integral problem.

Integral Evaluation

Integral evaluation often requires a blend of techniques, identities, and substitutions. In this exercise, evaluating the integral \( \int \frac{\mathrm{dx}}{1-\cos^4 \mathrm{x}} \) begins with recognizing applicable identities like the Pythagorean identity and using algebraic methods such as the difference of squares.
Successfully evaluating the integral means matching it with a known form, in this case derived from trigonometric functions. Here, the challenge is not just performing calculations, but also validating results against given options to confirm accuracy. This approach underscores the importance of checking work and ensuring each step ladders correctly towards a final, recognizable result, like the provided expression \(-(1/2)\cot{x}+\operatorname{Atan}^{-1}[f(x)]+c\).

• Knowing how to rewrite complex terms into integrable components is vital.
• Pair the mathematical techniques with problem-solving intuition.
• The application of multiple mathematical principles often dances seamlessly in integral evaluations.

Even when a solution is found, confirming consistency between initially complex integrals and resulting expressions remains crucial in calculus.