Problem 72 - Calculus Single Variable

Question

Calculate \(\int_{0}^{\pi / 2} 4 x \cos ^{2}(x) d x\).

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{\pi^2}{4}\).

Step 1: Identify and Simplify the Integrand

The integral given is \(\int_{0}^{\pi / 2} 4x \cos^{2}(x)\, dx\). Notice that \(\cos^2(x)\) can be rewritten using the identity \(\cos^2(x) = \frac{1 + \cos(2x)}{2}\). Substitute this into the integral, resulting in \(\int_{0}^{\pi / 2} 4x \frac{1 + \cos(2x)}{2}\, dx\). Simplify this to \(2 \int_{0}^{\pi / 2} x \left(1 + \cos(2x)\right)\, dx\).

Step 2: Distribute and Split the Integral

Distribute the \(x\) inside the integral to separate terms: \(2 \int_{0}^{\pi / 2} x \, dx + 2 \int_{0}^{\pi / 2} x \cos(2x)\, dx\). This turns the original integral into the sum of two simpler integrals: \(2 \int_{0}^{\pi / 2} x \, dx + 2 \int_{0}^{\pi / 2} x \cos(2x)\, dx\).

Step 3: Calculate the First Integral

Calculate \(2 \int_{0}^{\pi / 2} x \, dx\). The antiderivative of \(x\) is \(\frac{x^2}{2}\). Evaluate the definite integral as follows: \(2 \left[\frac{x^2}{2}\right]_{0}^{\pi / 2} = x^2\bigg|_{0}^{\pi / 2} = \left(\frac{\pi^2}{4}\right) - 0 = \frac{\pi^2}{4}\). Thus, the result is \(\frac{\pi^2}{4}\).

Step 4: Set Up the Second Integral with Integration by Parts

The second integral is \(2 \int_{0}^{\pi / 2} x \cos(2x)\, dx\). Use integration by parts, where \(u = x\) and \(dv = \cos(2x)\, dx\). Then, \(du = dx\) and \(v = \frac{1}{2} \sin(2x)\) after integrating \(dv\). Integration by parts formula is \(\int u \, dv = uv - \int v \, du\).

Step 5: Apply Integration by Parts

Calculate \(uv - \int v \, du\) for the second integral: \(2 \left[x \frac{1}{2} \sin(2x) - \int \frac{1}{2} \sin(2x) \, dx\right]_{0}^{\pi / 2}\). This simplifies to \(\left[x \frac{1}{2} \sin(2x)\right]_{0}^{\pi/2} - \int \frac{1}{2} \sin(2x) \, dx\).

Step 6: Evaluate the Boundary Terms

Evaluate \(\left[x \frac{1}{2} \sin(2x)\right]_{0}^{\pi/2}\): when \(x = \pi/2\), the expression is \(\frac{\pi}{4} \times 0 = 0\); when \(x = 0\), the expression is 0. So, the boundary terms sum to 0. Move on to compute the remaining integral \(\int \frac{1}{2} \sin(2x)\, dx\).

Step 7: Evaluate the Remaining Integral

The integral \(\int \frac{1}{2} \sin(2x)\, dx\) can be solved by substituting \(\frac{1}{2} \times -\frac{1}{2} \cos(2x)\). This evaluates to \(-\frac{1}{4} \cos(2x)\). Thus, \(\left[-\frac{1}{4} \cos(2x)\right]_{0}^{\pi/2}\).

Step 8: Combine both Results

Calculate \(-\frac{1}{4} \cos(2x)\) from 0 to \(\pi/2\): at \(\pi/2\), \(-\frac{1}{4} \times 1 = -\frac{1}{4}\); at 0, \(-\frac{1}{4} \times 1 = -\frac{1}{4}\). Thus, the result is \(-\frac{1}{4} - (-\frac{1}{4}) = 0\). Add this to the contribution of 0 from earlier. The second integral result is therefore 0.

Step 9: Sum Total Results

The first integral was \(\frac{\pi^2}{4}\) and the second contributed 0. Thus, the total integral is \(\frac{\pi^2}{4} + 0 = \frac{\pi^2}{4}\).

Key Concepts

Definite Integrals

Definite integrals help us find the area under a curve between two specified points on the x-axis. They are incredibly useful in various fields such as physics, engineering, and economics to calculate things like total distance, work done by a force, or cumulative cost.
The notation for definite integrals is \( \int_{a}^{b} f(x) \, dx \), where:

• \( a \) and \( b \) are the limits of integration, representing the interval over which we are integrating,
• \( f(x) \) is the function we are integrating, also known as the integrand,
• The result is a number that gives us the net area under the curve of \( f(x) \) from \( x = a \) to \( x = b \).

When calculating a definite integral, we first find the antiderivative (or indefinite integral) of the function. Then, we evaluate it at the upper and lower limits and subtract the lower value from the upper value to find the definite integral.

Integration by Parts

Integration by parts is a technique used to integrate products of functions. This method is essential when integrating products that do not simplify easily. It's based on the product rule for differentiation and is expressed with the formula \( \int u \, dv = uv - \int v \, du \).
Here's a step-by-step on how to use it effectively:

• Identify parts of the integrand to set as \( u \) and \( dv \). A common mnemonic for choosing \( u \) is LIPET: Logarithmic, Inverse trigonometric, Polynomial, Exponential, Trigonometric. Choose \( u \) from the listed categories, giving preference in that order.
• Differentiating \( u \) gives \( du \), while integrating \( dv \) results in \( v \).
• Insert these into the integration by parts formula and simplify.

This method is ideal for integrals that combine polynomials with trigonometric, exponential, or logarithmic functions. Practice makes perfect when applying this technique!

Trigonometric Identities

Trigonometric identities are equations involving trigonometric functions that are true for every value of the occurring variables. They are instrumental in simplifying integrands during integration.
Common trigonometric identities include:

• \( \sin^2(x) + \cos^2(x) = 1 \)
• \( 1 + \tan^2(x) = \sec^2(x) \)
• \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \)

In the given problem, the identity \( \cos^2(x) = \frac{1 + \cos(2x)}{2} \) is used to transform the integral into a more manageable form. By substituting identities, especially those involving double angles, it becomes easier to integrate otherwise complex expressions. Knowing when and how to apply these identities is key to mastering calculus problems.