\(\int\left[(\log \mathrm{x}) /(1+\log \mathrm{x})^{2}\right] \mathrm{d} \mathrm{x}=\underline{ }+\mathrm{c}\) (a) \([\mathrm{x} /(1+\log \mathrm{x})]\) (b) \(x(1+\log x)\) (c) \([\mathrm{x} /(\log \mathrm{x})]\) (d) \(x \log x+x^{-1}\)

Calculating an indefinite integral, also known as antiderivative, involves finding a function that represents all the antiderivatives of a given function. The result includes an arbitrary constant, often denoted as 'c', to represent the family of all possible solutions. The notation for an indefinite integral is shown as \( \int f(x) \, dx \), where \( f(x) \) is the integrand and \( dx \) indicates that the integration is with respect to the variable \( x \).

For example, the indefinite integral of \( x^2 \) is \( \frac{1}{3}x^3 + c \) because the derivative of \( \frac{1}{3}x^3 \) is \( x^2 \). This process is fundamental in calculus, as it allows us to determine quantities like area under a curve and the antiderivative of velocity, which is position.

Logarithmic integration involves integrating functions with logarithms. A key fact is that the integral of \( 1/x \) is \( \log|x| + c \), where \( c \) is the integration constant. When dealing with more complex logarithmic expressions, substitution usually simplifies the integral into a form where this relationship can be applied. This technique is especially useful when the argument of a logarithmic function appears in the numerator of a fraction.

For instance, the indefinite integral of \( (\log x)/x \) can directly be integrated to \( (1/2)\log^2|x| + c \) by recognizing the presence of the function and its derivative. This reflects the inseparable relationship between logarithmic functions and their integrals in calculus.

U-Substitution is a technique designed to simplify integration by changing the variable of integration into something more manageable. It's particularly useful when dealing with compositions of functions, like a logarithm under a power, as seen in the given exercise. The general idea is to let \( u \) equal a part of the integral that makes the derivative simpler, often to cancel out other parts of the integrand when \( dx \) is substituted.

To execute u-substitution, we differentiate \( u \) with respect to \( x \), solve for \( dx \), and substitute all instances of \( x \) with \( u \) in the integrand. After integration, we 'back-substitute' \( u \) with the original variable's terms. This method can significantly reduce the complexity of an integration problem and is a staple in any calculus toolkit.

A variety of integration techniques exist to tackle different kinds of integrals. Besides u-substitution, there are methods such as integration by parts, partial fractions, and trigonometric integration, among others. Knowing when to apply a specific technique comes with practice and understanding the form of the function you are integrating.

For instance, integration by parts is suitable for products of functions, and partial fractions work well with rational functions that can be decomposed. Trigonometric integration helps with functions involving sine and cosine functions. Mastering these techniques allows you to integrate a wide array of functions, enabling you to solve more complex calculus problems with confidence.