Problem 53 - Thomas Calculus

Question

Evaluate the integrals. \(\int_{0}^{\pi / 2} 7^{\cos t} \sin t d t\)

Step-by-Step Solution

Answer

The integral evaluates to \( \frac{6}{\ln 7} \).

Step 1: Recognizing the Integral Form

The given integral is \( \int_{0}^{\pi / 2} 7^{\cos t} \sin t \, dt \). This type of integral often suggests a substitution might simplify the problem.

Step 2: Choosing a Substitution

Let's use the substitution \( u = \cos t \). Thus, the derivative \( du = -\sin t \, dt \), or \( \sin t \, dt = -du \). This substitution changes the bounds as well: when \( t = 0, u = 1 \) and when \( t = \pi/2, u = 0 \).

Step 3: Substituting and Changing the Limits

Substitute \( u = \cos t \) into the integral to get \( \int_{1}^{0} 7^u (-du) \). The negative sign can be used to swap the limits giving \( \int_{0}^{1} 7^u \, du \).

Step 4: Integrating the New Expression

The integral \( \int 7^u \, du \) is equal to \( \frac{7^u}{\ln 7} + C \). So, \( \int_{0}^{1} 7^u \, du = \left[ \frac{7^u}{\ln 7} \right]_{0}^{1} \).

Step 5: Evaluating the Definite Integral

Compute \( \left[ \frac{7^u}{\ln 7} \right]_{0}^{1} = \frac{7^1}{\ln 7} - \frac{7^0}{\ln 7} = \frac{7}{\ln 7} - \frac{1}{\ln 7} = \frac{7 - 1}{\ln 7} = \frac{6}{\ln 7} \).

Key Concepts

Definite Integrals

Definite integrals are used in calculus to calculate the accumulated value of a function over a given interval. This involves computing the area under the curve of the function between two specific points on the x-axis. For the integral \( \int_{a}^{b} f(x) \, dx \), \( a \) and \( b \) are the limits of integration, indicating the start and end points of the interval respectively.

Instead of producing another function as the result, like indefinite integrals do, definite integrals provide a single numerical value. This makes them very useful for solving problems in physics and engineering where measurements over a specific distance or period are needed.

Definite integrals are evaluated using the Fundamental Theorem of Calculus which states:

• If \( F(x) \) is an antiderivative of \( f(x) \), then
• \( \int_{a}^{b} f(x) \, dx = F(b) - F(a) \).

It's important to switch the limits of integration correctly if the substitution involves reversing the order, as seen in the example, where the substitution resulted in the limits changing from \( 0 \) to \( \pi/2 \) to \( 1 \) to \( 0 \). Switching them gives the correct positive integral.

Substitution Method

The substitution method is a popular technique used in calculus to simplify complex integrals. This technique often comes into play when an integral contains composite functions, or when a direct algebraic integration seems difficult. The method involves substituting a part of the integral with a new variable, which facilitates easier integration.

The process typically involves:

• Identifying a substitution \( u = g(x) \) which simplifies the integral.
• Finding the differential \( du = g'(x) \, dx \), helping to express \( dx \) in terms of \( du \).
• Changing the limits of integration as per the new variable \( u \).
• Evaluating the simpler integral in terms of \( u \).

In our example, \( u = \cos t \) was chosen which transformed the integral into an easier form \( \int_{0}^{1} 7^u \, du \). Always remember to revert to the original variable if needed, although in definite integrals, the substitution often allows completing the problem in the new variable context.

Integration Techniques

There are various integration techniques in calculus, each suited for different types of integrals. Two of these were crucial in solving the given exercise: the substitution method and properties of exponential functions in integration.

Each technique aims to simplify integration by either breaking it down into simpler parts or transforming it into a form that's more straightforward to integrate:

• Substitution: Makes a difficult integral easier by changing variables.
• Exponential Integration: Involves integrations of functions like \( a^x \) where the integral is given by \( \frac{a^x}{\ln a} + C \) for an indefinite integral.

In our example, once we had \( \int_{0}^{1} 7^u \, du \), we used the property of exponentials to integrate, resulting in \( \frac{7^u}{\ln 7} \). Understanding when and how to apply these techniques ensures accurate solutions to calculus problems.