Problem 44 - Thomas Calculus in SI Units

Question

Evaluate the integrals in Exercises \(29-50.\) $$\int_{\pi / 4}^{\pi / 2}\left(1+e^{\cot \theta}\right) \csc ^{2} \theta d \theta$$

Step-by-Step Solution

Answer

The integral evaluates to \( e \).

Step 1: Substitution

To solve the integral \( \int_{\pi / 4}^{\pi / 2} (1 + e^{\cot \theta}) \csc^2 \theta \, d\theta \), we start by applying a substitution. Let \( u = \cot \theta \). Then \( \frac{du}{d\theta} = -\csc^2 \theta \), meaning \( du = -\csc^2 \theta \, d\theta \). When \( \theta = \pi/4 \), \( u = 1 \), and when \( \theta = \pi/2 \), \( u = 0 \). Thus, we rewrite the integral in terms of \( u \):\[ -\int_{1}^{0} (1 + e^u) \, du \]

Step 2: Flip Limits and Simplify

Reverse the limits of integration to switch the order, which requires changing the sign of the integral:\[ \int_{0}^{1} (1 + e^u) \, du \]

Step 3: Integrate with Respect to u

Now, we perform the integration:\[ \int (1 + e^u) \, du = \int 1 \, du + \int e^u \, du = u + e^u + C \]Evaluate this from 0 to 1:\[ \left[ u + e^u \right]_{0}^{1} \]

Step 4: Evaluate the Integral

Substitute the limits back into the integrated function:\[ \left(1 + e^1\right) - \left(0 + e^0\right) = (1 + e) - (0 + 1) = e \]

Step 5: Conclusion

Therefore, the value of the integral is \( e \).

Key Concepts

Definite Integrals

In calculus, a definite integral is a type of integral with upper and lower limits. These limits indicate the range over which the integration is performed.
This integral represents the total accumulation of quantities, such as area under a curve, over a specific interval.
The definite integral of a function, say \( f(x) \), between the limits \( a \) and \( b \) is denoted as

• \( \int_a^b f(x) \, dx \)

When computing a definite integral, we use the Fundamental Theorem of Calculus.
This connects the concept of an antiderivative with the accumulation function and involves evaluating the antiderivative at the given limits and subtracting the results. Once you find the antiderivative \( F(x) \), you calculate

• \( F(b) - F(a) \)

This provides a single number representing the totality of the function over the specified interval.

Trigonometric Substitution

Trigonometric substitution is a strategy used in integral calculus to simplify integrals, especially those involving square roots or quadratic expressions.
We replace variables with trigonometric functions to take advantage of Pythagorean identities.
This approach is most commonly applied when integrals contain expressions like \(a^2 - x^2\), \(a^2 + x^2\), or \(x^2 - a^2\).

• For expressions involving \(a^2 - x^2\), set \(x = a \sin \theta\).
• For \(a^2 + x^2\), use \(x = a \tan \theta\).
• With \(x^2 - a^2\), substitute \(x = a \sec \theta\).

In the discussed exercise, trigonometric substitution helps manage complexities arising from terms like \(\csc^2 \theta\).
Substituting \(u = \cot \theta\) simplifies the integration process by converting difficult trigonometric integrals into more manageable forms.

Integration by Substitution

Integration by substitution is a method aimed at simplifying integrals by changing variables.
It's akin to the reverse process of the chain rule in differentiation.
The main idea is to identify a part of the integral that can be represented as a new variable \( u \).

• With this method, you usually let \( u \) be a function of the original variable, say \( \theta \), and express \( du \) accordingly.

In the solution process of the given exercise, we chose \( u = \cot \theta \) making differentiation straightforward as \( du = -\csc^2 \theta \, d\theta \).
This substitution simplifies the expression, allowing us to rewrite the integral in a way that is easier to evaluate.
After substitution, all instances of the original variable are removed by expressing them in terms of \( u \), without mixing with the original variable, which allows for cleaner integration.