Question

\((\pi / 2) \int_{0} \sqrt{(\sec x+1) d x}\) is equal to \(\ldots \ldots\) (a) 0 (b) \((\pi / 4)\) (c) \((\pi / 2)\) (d) \(\pi\)

Short answer

(c) \((\pi / 2)\)

Step-by-step solution

Perform the trigonometric substitution

Substitute the given trigonometric identity: \(\sec{x} = \frac{1}{\cos{x}}\) in the integrand, so we get: \((\pi / 2) \int_{0} \sqrt{(\frac{1}{\cos{x}} + 1) dx}\)

Simplify the integrand

Now, let's simplify the integrand more by combining fractions inside the square root: \((\pi / 2) \int_{0} \sqrt{(\frac{1+\cos{x}}{\cos{x}}) dx}\)

Use the Pythagorean identity

We can further simplify the integrand by noticing that the numerator can be written in terms of sine using the Pythagorean identity \(1 - \sin^2{x} = \cos^2{x}\). So, the integrand becomes: \((\pi / 2) \int_{0} \sqrt{(\frac{2\sin^2{x/2}}{\cos{x}}) dx}\)

Cancel terms in the integrand

Now divide both the numerator and denominator by \(\cos{x}\). For this, we need to remember the double angle identity \(\sin{x} = 2\sin(x/2)\cos(x/2)\). After cancelation we get: \((\pi / 2) \int_{0} \sqrt{(4\sin^2{x/2}\cos^2{x/2}) dx}\)

Perform integration

Upon integrating the simplified integral, the result is: \((\pi / 2) \int_{0} (2\sin{x/2}\cos{x/2}) dx\) Using substitution, let \(u = x/2\), so \(du = dx/2\), and the bounds of integration become \(0 /2 = 0\) and \(\pi / 4\): \(2(\pi / 2) \int_{0}^{\pi / 4} (\sin{u}\cos{u}) (2 du)\) Simplifying further, we have: \(2\pi \int_{0}^{\pi / 4} (\sin{u}\cos{u}) d u\)

Evaluate the definite integral

Now integrate and evaluate \(2\pi [(\frac{1}{2} \sin^2{u})|_0^{\pi / 4}\) \(2\pi (\frac{1}{2} \sin^2{(\pi/4)} - \frac{1}{2} \sin^2{(0)})\) Plugging the values for the sine functions, we get: \(2\pi (\frac{1}{2}(1/2) - 0)\), Which simplifies to: \((\pi / 2)\) Thus, the answer is (c) \((\pi / 2)\).

Key concepts

Trigonometric Substitution

Trigonometric substitution is a valuable tool when dealing with integrals that involve expressions with roots. It involves replacing a variable with a trigonometric function, allowing for a simplification that turns complex algebraic forms into more approachable trigonometric forms. In this exercise, the trigonometric identity \(\sec{x} = \frac{1}{\cos{x}}\) plays a vital role. This substitution transforms the integrand into an expression suitable for integration techniques. When using trigonometric substitution, always aim to make your integrals resemble basic trigonometric identities, which are easier to integrate. This method is particularly efficient when the integral involves square roots.

Pythagorean Identity

The Pythagorean identity is a fundamental relationship in trigonometry, expressed as \(1 - \sin^2{x} = \cos^2{x}\). In this exercise, this identity is utilized to transform the integrand into a simpler form suitable for integration. By converting expressions involving higher powers of sine and cosine into sums or differences that feature squared terms, the Pythagorean identity serves as a bridge to simpler algebraic structures. For instance, adjusting the denominator from \(1 + \sin{x}\) to \(2\sin^2{x/2}\) using this identity helps streamline the calculation. When simplifying integrals, always check if the Pythagorean identity can ease the process, saving time and providing clarity.

Definite Integral Evaluation

Evaluating definite integrals involves finding the net area under a curve within a specific interval. This process requires not only setting up the integral properly but also ensuring that all bounds reflect the transformations made during substitution. In this exercise, after transforming the integral through trigonometric substitution and simplifications, we finally compute from the lower limit of 0 to the upper limit of \(\pi/2\). By correctly handling substitutions and changes in limits, such as using \(u = x/2\), we align the evaluation procedure with the transformations applied. This precision ensures accurate computations and meaningful results.

Integration Techniques

A variety of integration techniques can be applied to solve integrals, often involving multiple steps or a combination of methods. Here, integrating by parts and substitutions like \(u = x/2\) prove vital. This substitution simplifies the bounds and the integrand itself, making it easier to integrate. Such approaches highlight the importance of strategic planning in integration, where transformations create simpler integrals that are easier to handle. Each technique presents a strategy to unravel complicated expressions into more straightforward calculations, demonstrating the toolbox approach to solving integrals. Understanding when and how to deploy these techniques is crucial in efficiently solving definite integrals.