Problem 62 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int_{\pi / 6}^{\pi / 4} \frac{\csc ^{2} x d x}{1+(\cot x)^{2}}$$

Step-by-Step Solution

Answer

The value of the integral is \( \frac{\pi}{12} \).

Step 1: Simplify the Integral

To simplify the integral, we first note that \(1 + (\cot x)^2\) can be rewritten using the Pythagorean identity for cotangent:\[1 + \cot^2 x = \csc^2 x\]Substituting this into the integral, we get:\[\int_{\pi/6}^{\pi/4} \frac{\csc^2 x}{\csc^2 x} \, dx = \int_{\pi/6}^{\pi/4} 1 \, dx\]

Step 2: Evaluate the Simplified Integral

Now the problem simplifies to evaluating:\[\int_{\pi/6}^{\pi/4} 1 \, dx\]The integral of 1 with respect to \(x\) is simply \(x\). So we evaluate it as:\[\left[ x \right]_{\pi/6}^{\pi/4} = \left(\frac{\pi}{4}\right) - \left(\frac{\pi}{6}\right)\]

Step 3: Calculate the Definite Integral

Substitute the upper and lower limits into the evaluated expression:\[\frac{\pi}{4} - \frac{\pi}{6}\]To compute this, we need a common denominator:\[\frac{\pi}{4} - \frac{\pi}{6} = \frac{3\pi}{12} - \frac{2\pi}{12} = \frac{\pi}{12}\]

Step 4: Conclude the Result

The calculation finds that the value of the integral is:\[\frac{\pi}{12}\]

Key Concepts

Integral Simplification

When evaluating an integral, simplification can make the process much easier. In this exercise, we start by simplifying a complex-looking expression inside the integral using a trigonometric identity. The original integral was:\[ \int_{\pi / 6}^{\pi / 4} \frac{\csc^2 x}{1+(\cot x)^2} \, dx \]The trick here is to use a well-known identity: \(1 + \cot^2 x = \csc^2 x\). This helps transform the integral into a simpler form. By understanding and applying this identity, we can rewrite the problem as:\[ \int_{\pi / 6}^{\pi / 4} \frac{\csc^2 x}{\csc^2 x} \, dx = \int_{\pi/6}^{\pi/4} 1 \, dx \]This simplification is key. Once the integral simplifies to 1, evaluating it becomes straightforward and allows us to focus on the bounds of integration.

Definite Integrals

Definite integrals are used to calculate the accumulation of quantities and often represent area under a curve between two bounds. Here, we dealt with:\[ \int_{\pi/6}^{\pi/4} 1 \, dx \]Evaluating this definite integral involves finding the antiderivative first, which in this case is very simple as the antiderivative of 1 with respect to \(x\) is just \(x\).To solve the definite integral, plug in the upper limit and subtract the lower limit from it:

• Upper limit: \( \frac{\pi}{4} \)
• Lower limit: \( \frac{\pi}{6} \)

Finally, the computation becomes:\[ \left[ x \right]_{\pi/6}^{\pi/4} = \left(\frac{\pi}{4}\right) - \left(\frac{\pi}{6}\right) \]This straightforward calculation shows the power of definite integrals in finding differences over a specific interval.

Trigonometric Identities

Trigonometric identities play a critical role in calculus, especially as a tool for simplification. In this particular problem, the Pythagorean identity for cotangent and cosecant was pivotal:\[1 + \cot^2 x = \csc^2 x\]This identity allows us to realize that the components within the integral can be broken down or rewritten in a simpler form. In this problem, using the identity enabled re-writing \(1 + (\cot x)^2\) as \(\csc^2 x\), thereby greatly simplifying the original problem from a rational expression into just a constant:\[ \frac{\csc^2 x}{\csc^2 x} = 1 \]Knowledge of these identities allows for quick transformations and simplifications during integration. These transformations often reveal more manageable expressions, making the entire problem-solving process much more efficient and less error-prone. Always keep a handy list of common trigonometric identities, as they can be game-changers in solving calculus problems.