Question

\(\int\left[(1+\mathrm{x}) /\left(1+{ }^{3} \sqrt{\mathrm{x}}\right)\right] \mathrm{d} \mathrm{x}=\underline{\mathrm{x}}=\mathrm{c}\) (a) \((3 / 5) \mathrm{x}^{5 / 3}-(3 / 4) \mathrm{x}^{4 / 3}-\mathrm{x}\) (b) \((3 / 5) x^{5 / 3}-(3 / 4) x^{4 / 3}+x\) (c) \((3 / 5) \mathrm{x}^{5 / 3}+(3 / 4) \mathrm{x}^{4 / 3}-\mathrm{x}\) (b) \((3 / 5) \mathrm{x}^{5 / 3}+(3 / 4) \mathrm{x}^{4 / 3}-\mathrm{x}\)

Short answer

None of the given choices match the correct integral, which is \(-\frac{1}{3x^{1/3}}+\frac{x^{1/3}}{3} + C\).

Step-by-step solution

Identify the Function and the Variable

The function that needs integrating is \(\frac{1+x}{1+x^{1/3}}\) and the variable to be integrated with respect to is \(x\).

Perform Substitution

Let's perform a substitution so that we can simplify the function before integration. Let \(u = x^{1/3}\), so \(u^3 = x\). Differentiating both sides with respect to x, we get \(3u^2\frac{du}{dx} = 1\), so \(\frac{du}{dx} = \frac{1}{3u^2}\). Now, substituting \(u\) in the function and replacing \(\mathrm{d}\mathrm{x}\) with \(\frac{dx}{du}\), we get: \(\int\left[\frac{1+u^3}{1+u}\right]\frac{1}{3u^2}\mathrm{d}\mathrm{u}\).

Simplify and Integrate

Simplify the function inside the integral: \(\int\left[\frac{1+u^3}{1+u}\right]\frac{1}{3u^2}\mathrm{d}\mathrm{u}= \int\left[\frac{1+u^2}{3u^2}\right]\mathrm{d}\mathrm{u}\) Now let's integrate: \(\int\left[\frac{1+u^2}{3u^2}\right]\mathrm{d}\mathrm{u} = \frac{1}{3}\int\left[\frac{1}{u^2}+\frac{u^2}{u^2}\right]\mathrm{d}\mathrm{u} = \frac{1}{3}\int\left[\frac{1}{u^2}+1\right]\mathrm{d}\mathrm{u}\) Integrate each term in the integral: \(\frac{1}{3}\left[-\frac{1}{u} + u + C\right] = -\frac{1}{3u}+\frac{u}{3} + C\)

Substituting the original variable

Substitute the original variable \(x\) back in place of \(u\): \(-\frac{1}{3x^{1/3}}+\frac{x^{1/3}}{3} + C\)

Compare to the Given Choices

Now let's compare our integral result with the given choices: (a) \(\frac{3}{5}x^{5/3} - \frac{3}{4}x^{4/3} - x\) (b) \(\frac{3}{5}x^{5/3} - \frac{3}{4}x^{4/3} + x\) (c) \(\frac{3}{5}x^{5/3} + \frac{3}{4}x^{4/3} - x\) (d) \(\frac{3}{5}x^{5/3} + \frac{3}{4}x^{4/3} + x\) None of the given choices match our solution. It is possible that there is a mistake in the exercise, or the provided choices are incorrect.

Key concepts

Indefinite Integral

The term 'indefinite integral' is used interchangeably with 'antiderivative' and represents the family of all functions that could be the derivative of the original function. In essence, when we carry out an indefinite integration, like in the exercise \( \int\left[(1+\mathrm{x}) /(1+x^{1/3})\right]\mathrm{d} \mathrm{x} \), we're seeking a function whose derivative gives us the integrand \( (1+x)/(1+x^{1/3}) \).

Every indefinite integral includes an arbitrary constant, often denoted as \(C\), because the derivative of a constant is zero. This means you can add any constant to the function and still have the same derivative, hence why the constant of integration is essential.

Substitution Method in Integration

The substitution method simplifies integrals by changing the variable of integration to something more manageable. Commonly, this method is analogous to the 'u-substitution' used here, where a part of the integrand is substituted with \(u\), making the equation easier to integrate.

When implementing this method, it's crucial not only to replace the variable but also to convert the differential—here, \(dx\) was changed to \(\frac{du}{3u^2}\) after the substitution \(u = x^{1/3}\). The trick lies in choosing a substitution that simplifies the integrand effectively, which sometimes requires a bit of practice and intuition.

Integrating Rational Functions

Rational functions are quotients of polynomials. They can often be integrated by means of various techniques like partial fractions, polynomial long division, or substitutions which we've seen in this problem. In the provided exercise, the substitution method was used to reach a simpler rational function in the form of \(\frac{1+u^2}{3u^2}\), which is then easier to integrate.

After simplification, the function inside the integral had the form of a sum of simpler fractions that could be readily integrated term by term, showcasing an approach to integrating more complex rational functions.

JEE Maths Problems

Mathematics problems in the Joint Entrance Examination (JEE), which is an engineering entrance assessment conducted in India, are known for their challenging nature and often require a strong grasp of concepts and the ability to apply various strategies like the substitution method. This exercise is an example of problems that could appear in such exams, where candidates must perform accurate integrations, perhaps with substitutions, to solve the question.

In JEE and other competitive exams, attention to detail is paramount, as a small mistake can lead to incorrect answers, as demonstrated in the step-by-step solution that did not match any of the provided choices, indicating a possible error in the exercise or options listed.