Problem 45 - Thomas Calculus

Question

Use a substitution to change the integral into one you can find in the table. Then evaluate the integral. \(\int \cot t \sqrt{1-\sin ^{2} t} d t, \quad 0< t< \pi / 2\)

Step-by-Step Solution

Answer

The integral evaluates to \( \ln(\csc t - \cot t) + \cos t + C \).

Step 1: Identify the Substitution

Observe that the integrand contains \( \sqrt{1 - \sin^2 t} \). We know that \( \cos^2 t = 1 - \sin^2 t \), so \( \sqrt{1 - \sin^2 t} = \cos t \). This means we can simplify the integral by substituting this expression into the integral.

Step 2: Substitute in the Integral

Replace \( \sqrt{1 - \sin^2 t} \) with \( \cos t \) in the integral, transforming it to \( \int \cot t \cos t \, dt \). Recognize that \( \cot t = \frac{\cos t}{\sin t} \), so the integral becomes \( \int \frac{\cos^2 t}{\sin t} \, dt \).

Step 3: Simplify the Expression

The integral simplifies to \( \int \frac{\cos^2 t}{\sin t} \, dt \). Recall that \( \cos^2 t = 1 - \sin^2 t \), and rewrite the integral as \( \int \frac{1 - \sin^2 t}{\sin t} \, dt \). This can be split into two separate integrals: \( \int \frac{1}{\sin t} \, dt - \int \sin t \, dt \).

Step 4: Evaluate the First Integral

The first integral, \( \int \frac{1}{\sin t} \, dt = \int \csc t \, dt \), is a standard integral. It evaluates to \( \ln | \csc t - \cot t | + C \).

Step 5: Evaluate the Second Integral

The second integral, \( \int \sin t \, dt \), is another standard integral. It evaluates to \( -\cos t + C \).

Step 6: Combine the Integrals

Combine the results from Steps 4 and 5 to express the original integral. The combined expression is: \( \ln | \csc t - \cot t | + \cos t + C \).

Step 7: Include Parameters and Consider the Interval

Since the problem provides bounds \( 0 < t < \pi/2 \), there is no need to include absolute values as all trigonometric functions are positive in this interval. Therefore, the final expression is \( \ln(\csc t - \cot t) + \cos t + C \).

Key Concepts

Trigonometric Identities

Trigonometric identities are mathematical equations involving trigonometric functions that hold true for all values of the involved variables. In our exercise, the crucial identity is that for any angle \( t \), \( \cos^2 t + \sin^2 t = 1 \). This Pythagorean identity helps simplify expressions when dealing with integrals. By rearranging the identity, we have \( \cos^2 t = 1 - \sin^2 t \). This allows us to express \( \sqrt{1 - \sin^2 t} \) as \( \cos t \), making the integral much simpler to solve. Using identities like this is vital when you want to transform a complex trigonometric integral into a more manageable form.

Trigonometric Integrals

Trigonometric integrals involve the integration of trigonometric functions. They often require using identities or transformations to evaluate. In this exercise, we dealt with integrals that initially seemed challenging due to the combination of expressions, such as \( \int \cot t \sqrt{1-\sin^2 t} \, dt \). A good strategy is to simplify the integral using known trigonometric identities, which can break down complex expressions into standard integrals. For example, once we rewrite the integral in terms of \( \cos t \) using \( \cos t = \sqrt{1 - \sin^2 t} \), we can further transform it to \( \int \frac{\cos^2 t}{\sin t} \, dt \), leading to basic integrals that are easier to handle.Standard integrals for trigonometric functions, such as \( \int \csc t \, dt \) and \( \int \sin t \, dt \), are solved based on known patterns. These are often kept in integration tables for reference.

Substitution Method

The substitution method is a powerful technique used in calculus for solving integrals that are not immediately straightforward. It involves changing variables to simplify the integration process. For this exercise, the key substitution was recognizing that \( \cos^2 t = 1 - \sin^2 t \), which allowed us to replace \( \sqrt{1 - \sin^2 t} \) with \( \cos t \). This transformation simplified an initially daunting expression into a more ordinary form. Once transformed, the integral \( \int \frac{\cos^2 t}{\sin t} \, dt \) can be split into simpler parts like \( \int \frac{1}{\sin t} \, dt \) and \( \int \sin t \, dt \). The substitution reduces the complexity by turning an expression into a more recognizable pattern or into a form included in integration tables.The trick of the substitution method is choosing the right substitution that makes the integral solvable by either simplifying the integrand or by fitting it into a standard form.