Question

\(\int[\sin (\log x)+\cos (\log x)] d x=\underline{ }+c\) (a) \(\sin (\log x)+\cos (\log x)\) (b) \(+x \sin (\log x)\) (c) \(-x \cos (\log x)\) (d) \(\sin (\log x)+\cos (\log x)\)

Short answer

The correct short answer not provided in the options is: \(2x\cos(\log x) + C\)

Step-by-step solution

Identify the parts for integration by parts

To solve the given integral, we will use integration by parts, where we will consider the following parts: \(u = \sin(\log x) + \cos(\log x)\) and \(dv = dx\) Now, we need to differentiate u and integrate dv: \(du = (\frac{1}{x} \cos(\log x) - \frac{1}{x} \sin(\log x))dx\) \(v = x\)

Apply integration by parts formula

Using the integration by parts formula, which is: \(\int u dv = uv - \int v du\) We can rewrite the given integral as: \(\int(\sin(\log x) + \cos(\log x))dx = (x[\sin(\log x) + \cos(\log x)]) - \int(x[\frac{1}{x} \cos(\log x) - \frac{1}{x} \sin(\log x)])dx\)

Simplify the integral

Now, we will simplify the expression before integrating it: \((x[\sin(\log x) + \cos(\log x)]) - \int([x\frac{1}{x} \cos(\log x)] - [x\frac{1}{x} \sin(\log x)])dx\) This simplifies to: \((x[\sin(\log x) + \cos(\log x)]) - \int(\cos(\log x) - \sin(\log x))dx\)

Integrate the simplified expression

Finally, let's integrate the simplified expression: \((x[\sin(\log x) + \cos(\log x)]) - [\int \cos(\log x)dx - \int \sin(\log x)dx]\) Now we integrate each term separately: \(\int \cos(\log x)dx = x\sin(\log x) + C_1\) \(\int \sin(\log x)dx = -x\cos(\log x) + C_2\) So, the final solution is: \((x[\sin(\log x) + \cos(\log x)]) - [x\sin(\log x) - (-x\cos(\log x))] + C\) \(= x[\sin(\log x) + \cos(\log x) - \sin(\log x) + \cos(\log x)] + C\) This can be further simplified to: \(x[2\cos(\log x)] + C\) Now compare this result with the given options: (a) \(\sin(\log x) + \cos(\log x)\) - Incorrect (b) \(+x \sin(\log x)\) - Incorrect (c) \(-x \cos(\log x)\) - Incorrect (d) \(\sin(\log x) + \cos(\log x)\) - Incorrect No given option matches the obtained result. Therefore, none of the given options (a), (b), (c), and (d) are correct for this integration problem.

Key concepts

Definite Integrals

Definite integrals are a fundamental concept in calculus that help us calculate the area under a curve between two points. This is in contrast to indefinite integrals, which provide the family of all antiderivatives of a function. The notation for a definite integral from point \(a\) to point \(b\) is given by: \[ \int_{a}^{b} f(x) \, dx \]where \(f(x)\) is the function being integrated, and \(a\) and \(b\) are the limits of integration.Definite integrals have many applications, such as finding areas, volumes, and work done by a force. To solve a definite integral, you usually:

• Find the antiderivative or indefinite integral of the function.
• Evaluate the antiderivative at the upper limit \(b\) and subtract its value at the lower limit \(a\).

Understanding definite integrals is crucial for mastering calculus, as they lay the groundwork for solving complex real-world problems. In our example exercise, while handling the integration process via integration by parts, we concentrated on the indefinite integral aspect first; the definite integrals come into play when adding limits later.

Logarithmic Functions

Logarithmic functions are the inverse of exponential functions. They are widely used in mathematics to describe quantities that grow or decay exponentially, such as sound intensity, pH levels, and earthquake magnitudes. The basic form of a logarithmic function is \(y = \log_b(x)\), where \(b\) is the base of the logarithm, and \(x\) is the argument.When solving integrals involving logarithmic functions, like \(\sin(\log x)\) or \(\cos(\log x)\), it is essential to understand how logarithmic transformations can simplify the integration process. For instance, using properties of logarithms such as \(\log(xy) = \log x + \log y\) can help in breaking down a complex integral.In the given problem, \(\log x\) is part of a composition inside trigonometric functions. This adds an additional layer of complexity but also opens pathways for solving the integral by use of substitution or integration by parts. Familiarity with the properties of logarithmic functions allows students to manipulate these functions effectively for integration.

Trigonometric Functions

Trigonometric functions are essential for understanding various phenomena in physics and engineering, such as waves and oscillations. Common trigonometric functions include sine, cosine, and tangent, often written as \(\sin(x)\), \(\cos(x)\), and \(\tan(x)\), respectively.These functions are periodic, meaning they repeat values in regular intervals. This property can simplify the integration process. For example, knowing that \(\sin(x+2\pi) = \sin(x)\) allows us to evaluate integrals over one period and apply it to multiple cycles.In the exercise task, both \(\sin(\log x)\) and \(\cos(\log x)\) are incorporated within the integral, showing how trigonometric functions can interact with other functions, such as logarithms. Trigonometric identities, such as \(\sin^2(x) + \cos^2(x) = 1\), are useful tools for simplifying complex trigonometric expressions in integrals, and can be leveraged to find simpler expressions or make substitutions.Mastering the integration of trigonometric functions is vital for solving real-world problems since they frequently model naturally occurring cycles and oscillations in various scientific fields.