Question

If \(\mathrm{f}(\mathrm{x})=\cos \mathrm{x}-\cos ^{2} \mathrm{x}+\cos ^{3} \mathrm{x}-\cos ^{4} \mathrm{x}+\underline{\mathrm{x}}+\) then \(\int f(x) d x=\) \(+c\) (a) \(\tan (\mathrm{x} / 2)\) (b) \(x+\tan (x / 2)\) (b) \(x-(1 / 2) \tan (x / 2)\) (d) \(x-\tan (x / 2)\)

Short answer

None of the given options match the integral we found.

Step-by-step solution

Break down the function

We have the following function: \(f(x) = \cos x - \cos^2 x + \cos^3 x - \cos^4 x + x\) Let's break this function into parts: \(f(x) = \cos x - \cos^2 x + \cos^3 x - \cos^4 x + x\)

Integrate each part

Now, we will integrate each part of the function: 1. \(\int (\cos x) dx = \sin x + C_1\) 2. \(\int (\cos^2 x) dx = \frac{x}{2} + \frac{1}{4} \sin 2x + C_2\) 3. \(\int (\cos^3 x) dx = \frac{1}{3}(\sin x -\frac{1}{2}\sin^3 x) + C_3\) 4. \(\int (\cos^4 x) dx = \frac{x}{2} + \frac{3}{8}\sin 2x - \frac{1}{32}\sin 4x + C_4\) 5. \(\int (x) dx = \frac{1}{2}x^2 + C_5\)

Add the integrals together

Now that we have found the integral of each part, we will add them together to find the integral of the entire function: \(\int f(x) dx = (\sin x + C_1) - (\frac{x}{2} + \frac{1}{4}\sin 2x + C_2) + (\frac{1}{3}(\sin x -\frac{1}{2}\sin^3 x) + C_3) - (\frac{x}{2} + \frac{3}{8}\sin 2x - \frac{1}{32}\sin 4x + C_4) + (\frac{1}{2}x^2 + C_5)\)

Simplify and combine constants

Finally, let's simplify the expression and combine all the constant terms: \(\int f(x) dx = \frac{3}{2}x^2 - \frac{1}{2}x + 2\sin x -\frac{1}{2}\sin^3 x - \frac{1}{4}\sin 2x - \frac{3}{8}\sin 2x + \frac{1}{32}\sin 4x + (C_1 - C_2 + C_3 - C_4 + C_5)\) By comparing this result with the given options, it seems that none of them matches our result exactly. Therefore, the integral is likely not feasible to be simplified into any of the given options (a, b, or d).

Key concepts

Trigonometric Integration

Trigonometric integration involves integrating functions that include trigonometric terms like sine, cosine, tangent, etc. Trigonometric functions often appear in integrals due to their cyclic nature. This type of integration can be tricky, as it often requires special techniques to solve.

In the problem, we firstly need to integrate trigonometric terms like \( \cos x \), \( \cos^2 x \), \( \cos^3 x \), and \( \cos^4 x \). Each component demands a deep understanding of trigonometric identities and integration techniques.

• For \( \int \cos x \, dx \), the integral reduces directly to \( \sin x \).
• For higher powers of cosine, such as \( \int \cos^2 x \, dx \), a common approach is to use trigonometric identities like \( \cos^2 x = \frac{1 + \cos 2x}{2} \) to convert them into more manageable forms.
• Similarly, \( \cos^3 x \) and \( \cos^4 x \) integrate by expressing them using identities or products to apply further integration techniques.

Understanding these fundamental trigonometric transformations is essential in solving integrals involving such terms.

Integration by Parts

Integration by parts is a technique derived from the product rule of differentiation. It's particularly useful when dealing with products of functions, where one function easily differentiates, and the other easily integrates. The formula is expressed as:

\[ \int u \, dv = uv - \int v \, du \]

In the given function \( f(x) \), integrating terms such as \( \cos^3 x \) may involve integration by parts if transformed sufficiently. Here, however, straightforward trigonometric identities help simplify the process before needing advanced techniques like integration by parts.

The method shines when polynomial and exponential or trigonometric functions are involved. In cases where direct integration is challenging, integration by parts provides a strategic path to simplifying the expression.

Indefinite Integral

Indefinite integrals are integrals without upper and lower limits, represented by the integral symbol followed by the function and the differential (\( \int f(x) \, dx \)).

An indefinite integral results in a general form solution with a constant \( C \), raised from the unknown constant resulting during integration. Here, our task is to find \( \int f(x) \, dx \).

This process involves integrating each term separately, then combining the results, while remembering that each integration may provide its own constant. The final step amalgamates these constants into a single constant \( C \). This reflects the general solution to the function’s antiderivative, covering the family of curves represented by adding \( C \) to result from any starting point on the curve of \( f(x) \).

Algebraic Simplification

Algebraic simplification is the process of rearranging and condensing expressions to their simplest forms. This step is crucial in making sense of complex integrals and arriving at an understandable solution.

When we integrate the function \( f(x) \), each part's integration resulted in a complex expression. Simplification means combining like terms, factoring out common factors, and removing redundancies.

• For instance, combining multiple \( \sin x \) terms or \( \cos x \) related terms into simpler components.
• The constant terms are aggregated into a single constant \( C \), representing the general solution's indifferent constant term where specifics can vary.

Simplification allows the direct comparison of our integral solutions with the options provided, aiming to format our answer to one of the available choices, even if, as in this case, the solution might remain unmatched. Understandably, not all complex integrals meet the concise form matching textbook options, reflecting integration’s sometimes open-ended nature.