Problem 54 - Thomas Calculus in SI Units

Question

Evaluate the integrals. $$\int \frac{\sec x d x}{\sqrt{\ln (\sec x+\tan x)}}$$

Step-by-Step Solution

Answer

The integral evaluates to \(\frac{2}{3}(\ln(\sec x + \tan x))^{3/2} + C\).

Step 1: Substitution

Start by setting a substitution to simplify the integral. Let \(u = \ln(\sec x + \tan x)\). This implies \(du = \frac{\sec x \tan x + \sec^2 x}{\sec x + \tan x} \, dx\).

Step 2: Simplify Differentials

Observe that \(du = \frac{\sec^2 x + \sec x \tan x}{\sec x + \tan x} \, dx\) which simplifies to \(du = \frac{\sec x}{\sqrt{\ln(\sec x + \tan x)}} \, dx\). This matches the form of the original integral giving us \(dx = \frac{\sqrt{u}}{\sec x} \, du\).

Step 3: Change of Variable in Integral

Use the substitution from Step 1 to change variables in the integral, transforming it into \( \int \frac{\sec x \, dx}{\sqrt{u}} = \int \sqrt{u} \, du \).

Step 4: Integrate with Respect to u

Integrate the simplified equation with respect to \(u\): \(\int \sqrt{u} \, du = \frac{2}{3}u^{3/2} + C\).

Step 5: Reverse the Substitution

Substitute back \(u = \ln(\sec x + \tan x)\) into the integral result to obtain the original variables: \(\frac{2}{3}\left(\ln(\sec x + \tan x)\right)^{3/2} + C\).

Key Concepts

Substitution Method

The substitution method is a powerful integration technique used to simplify complex integrals. The idea behind substitution is to transform a difficult integral into a simpler one by changing variables.

• First, identify a part of the integrand that can be substituted with a new variable.
• The original problem used the substitution: \[ u = \ln(\sec x + \tan x) \]which transforms the complex expression into a manageable form.
• By calculating the derivative, \[ du = \frac{\sec^2 x + \sec x \tan x}{\sec x + \tan x} \, dx \],we see that the substitution provides a convenient form where differentials can easily be replaced.

This method helps to set up the integral in a way that allows you to proceed with simpler integration techniques.

Change of Variables

The change of variables is crucial in manipulating integrals to a form that is easier to evaluate. In the process of integration, changing variables allows for a transformation that can significantly reduce complexity.

• Following the substitution, the integral is transformed using the substitution made, turning \[ \int \frac{\sec x \ dx}{\sqrt{\ln(\sec x + \tan x)}} \]into\[ \int \sqrt{u} \, du \].
• Notice how the integral becomes simpler after the change of variable is performed, showing the power of this technique.
• Change of variables often involves rewriting both the function and differential in terms of the new variable.

This approach significantly simplifies complex integrations by allowing you to perform the integral with respect to the new variable, dropping intricate original dependencies.

Integration Techniques

Several integration techniques can handle complex functions. Integration using substitution and change of variables, as demonstrated, are part of these techniques.

• In this problem, the transformation leads to an integral \[ \int \sqrt{u} \, du \] after substitution and change of variables, which is a standard integral.
• Solving \[ \int \sqrt{u} \, du \] involves standard techniques, leading to\[ \frac{2}{3}u^{3/2} + C \].
• After integration, reverse the substitution to express the result in terms of the original variable, giving:\[ \frac{2}{3}(\ln(\sec x + \tan x))^{3/2} + C \].

By combining these integration techniques, challenging integrals become more approachable and solvable.