Problem 68 - Thomas Calculus

Question

Evaluate the integrals in Exercises \(47-70\) $$ \int_{1}^{e^{\pi / 4}} \frac{4 d t}{t\left(1+\ln ^{2} t\right)} $$

Step-by-Step Solution

Answer

The evaluated integral is \( \pi \).

Step 1: Identify the Integral

The given integral is \( \int_{1}^{e^{\pi / 4}} \frac{4 \, dt}{t(1+\ln^{2} t)} \). We need to evaluate this definite integral from \( t = 1 \) to \( t = e^{\pi / 4} \).

Step 2: Choose a Substitution

To simplify the integration, use the substitution \( u = \ln t \). Then \( du = \frac{1}{t} dt \). The limits of integration change accordingly: when \( t = 1 \), \( u = 0 \), and when \( t = e^{\pi / 4} \), \( u = \frac{\pi}{4} \).

Step 3: Rewrite the Integral

Substitute \( u \) and \( du \) into the integral. The integral becomes:\[\int_{0}^{\pi/4} \frac{4 \, du}{1+u^2}\]

Step 4: Recognize the Integral Form

The integral \( \int \frac{du}{1+u^2} \) is known to be \( \arctan u + C \). Thus, we recognize that this integral is a standard form that results in the arctangent function.

Step 5: Evaluate the Integral

Evaluate the integral:\[\int_{0}^{\pi/4} \frac{4 \, du}{1+u^2} = 4 \left[ \arctan(u) \right]_{0}^{\pi/4}\]This becomes:\[4 \left( \arctan(\pi/4) - \arctan(0) \right)\]

Step 6: Calculate the Result

Calculate the arctangent values:- \( \arctan(\pi/4) \approx \frac{\pi}{4} \) (since \( \pi/4 \) is the angle whose tangent is \( \pi/4 \)).- \( \arctan(0) = 0 \).So, the result is:\[4 \left( \frac{\pi}{4} - 0 \right) = \pi\]

Key Concepts

Substitution Method

The substitution method is a powerful technique used in calculus to simplify the process of integration. By changing the variable of integration, it often transforms a complex integral into a more manageable form. In this exercise, we used the substitution \( u = \ln t \). This particular substitution is helpful because it simplifies the expression involving the natural logarithm. When employing substitution, always remember to:

• Choose a new variable, generally represented as \( u \), that simplifies the integral.
• Express \( dt \) in terms of \( du \) using the derivative of the substitution equation, in this case, \( du = \frac{1}{t} dt \).
• Adjust your integration limits according to the new variable.

By substituting, the initially complex-looking integral becomes a simple form that you can more easily solve.

Integration Limits

When performing definite integration, it is essential to adjust the integration limits to match the variable you are currently using. The original limits \( t = 1 \) to \( t = e^{\pi / 4} \) were suitable for the variable \( t \). However, after substitution, we switched to the variable \( u \).To update the limits:

• Substitute each original limit into the substitution formula. For example, when \( t = 1 \), substituting gives \( u = \ln(1) = 0 \).
• Do the same with the upper limit: when \( t = e^{\pi / 4} \), substituting gives \( u = \ln(e^{\pi / 4}) = \frac{\pi}{4} \).

By converting the limits of integration like this, we keep the integral consistent with the substitution variable, ensuring that the result of the integration will be accurate.

Arctangent Function

The arctangent, denoted as \( \arctan(x) \), is a fundamental function in calculus, specifically in integration. It is the inverse function of the tangent and is used when evaluating integrals of the form \( \int \frac{1}{1+u^2} \, du \). This is a standard integral that results in the function \( \arctan(u) \).Here’s how it works:

• The integral \( \int \frac{1}{1+u^2} \, du \) directly transforms into \( \arctan(u) + C \), where \( C \) is the constant of integration.
• This transformation is due to the derivative of the arctangent function: \( \frac{d}{du} (\arctan(u)) = \frac{1}{1+u^2} \).

In this exercise, after substitution, the integral was simplified to \( \int_{0}^{\pi/4} \frac{4 \, du}{1+u^2} \), allowing us to use the arctangent function for calculating the final result.

Change of Variable

A change of variable is a technique used to rewrite an integral in terms of a different variable, often simplifying the integral. In this exercise, changing from \( t \) to \( u \) made the integration much simpler.Considerations for a successful change of variable:

• Choose a substitution that simplifies the integrand to a more recognizable form.
• Express all parts of the integral in terms of the new variable, including converting the differential and adjusting integration limits.
• Look for standard integral forms that are easier to compute, such as the arctangent function in this example.

The change of variable performed in this problem turned a complicated expression into a basic form, making the computation straightforward and efficient.