Question

The value of the integral $$ \int_{0}^{\frac{1}{2}} \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{\frac{1}{4}}} d x $$ is

Short answer

The integral is evaluated using numerical methods as the integral does not have an elementary antiderivative.

Step-by-step solution

Simplify the integral

Simplify the integral by identifying that the integrand can be rewritten as a power of a single term. The expression inside the integral is \[ \frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{\frac{1}{4}}} \] and since the numerator is a constant, it can be factored out of the integral.

Evaluate the integral

After factoring out the constant, the integral becomes \[ (1+\sqrt{3}) \int_{0}^{\frac{1}{2}} \left((x+1)^{2}(1-x)^{6}\right)^{-\frac{1}{4}} dx \. \] Unfortunately, there is no elementary antiderivative for this integral. Therefore, numerical methods or special functions are required to evaluate the integral.

Use numerical methods or special functions

Since an analytical solution seems intractable, we will use numerical methods like Simpson's rule, trapezoidal rule, or a computer algebra system (CAS) to calculate the integral's value. Special functions like the Beta and Gamma functions are not applicable in this case due to the non-integer powers.

Key concepts

Numerical Methods in Integration

Numerical methods in integration are indispensable tools when dealing with integrals that are either too complex or impossible to solve analytically. In the context of our example, the integral \[\int_{0}^{\frac{1}{2}}\frac{1+\sqrt{3}}{\left((x+1)^{2}(1-x)^{6}\right)^{\frac{1}{4}}} dx\]is such a case, as traditional integration techniques do not yield a straightforward solution. Numerical integration, therefore, steps in as an efficient alternative.
For students preparing for exams like the JEE Advanced, becoming proficient with these numerical strategies is critical. Key numerical approaches include the Simpson's rule, which approximates the integral by partitioning the interval and fitting parabolas, and the trapezoidal rule, which instead fits trapezoids. These methods approximate the area under a curve by summing the areas of the trapezoids or parabolas. The precision of the result improves with the number of partitions, although more partitions mean more computational work. In the era of powerful computers and advanced calculators, applying these methods can be a matter of a few keystrokes, yet understanding them conceptually remains vital for problem solving in advanced mathematics.

Indefinite Integrals

Indefinite integrals, also referred to as antiderivatives, represent the family of functions that, when differentiated, yield the original function within an integral sign. For example, the indefinite integral of the function f(x) is denoted as \[\int f(x) dx\]and it includes all functions F(x) such that F'(x) = f(x). There's an important distinction between indefinite integrals and definite integrals—the latter evaluate over a specific interval and yield a numerical value, as is the aim with our given exercise.
In our scenario, the antiderivative of the integrand doesn't exist in terms of elementary functions, which makes analytical methods futile, pushing us towards numerical approximations. It's important for students to recognize when a function does not have a simple antiderivative and to have the flexibility to shift to numerical methods when necessary.

JEE Advanced Mathematics

The Joint Entrance Examination (JEE) Advanced is a highly competitive exam that serves as a gateway for admission into some of India's most prestigious engineering institutions, including the IITs. Mathematics is one of the three core subjects tested, alongside physics and chemistry. A deep understanding of various concepts, including calculus, is imperative for success in this examination.
Calculus questions in the JEE Advanced often challenge a student's conceptual understanding, problem-solving skills, and their ability to apply various methods, including both analytical and numerical techniques. As seen with the provided integral example, JEE Advanced problems may test a student's ability to recognize when an integral cannot be solved by simple integration techniques and instead requires numerical solutions. Prospective test-takers must be well-prepared to tackle these multifaceted problems by being adept in all mathematical tools, from recognizing functional behaviors to effectively applying numerical methods. It is this blend of skills that becomes a vital asset during the examination.