Problem 8 - 5 Steps to A 5: AP Calculus AB 2021

Question

Evaluate the following integrals in problems 1 to \(20 .\) No calculators are allowed. (However, you may use calculators to check your results.) $$\int \frac{\sin x}{\cos ^{3} x} d x$$

Step-by-Step Solution

Answer

Question: Evaluate the following integral: $$\int \frac{\sin x}{\cos^3 x} dx$$ Answer: The integral evaluates to: $$\frac{1}{2}\sec^2 x + C$$

Step 1: Identify the best substitution

Observe the integral and consider substituting \(u = \cos x\), given that the derivative of \(\cos x\) is \(-\sin x\). This substitution should simplify the integral.

Step 2: Perform substitution

Let \(u = \cos x\). Now, we'll find the derivative of \(u\) with respect to \(x\): $$\frac{du}{dx} = -\sin x$$ Then, obtain the relation between \(dx\) and \(du\): $$du = -\sin x dx$$ Now, substitute \(u\) and \(du\) into the integral: $$\int \frac{\sin x}{(\cos x)^3} dx = \int \frac{1}{u^3} (-du)$$

Step 3: Integrate the new function

Now we can integrate the function with respect to \(u\): $$-\int \frac{1}{u^3} du$$ Using the power rule for integration, we get: $$-\int u^{-3} du = -\frac{u^{-2}}{(-2)} + C$$

Step 4: Substitute back original function

Replace \(u\) with the original function, \(\cos x\): $$\frac{1}{2} \cdot \frac{1}{(\cos x)^2} + C$$

Step 5: Simplify the result

Using the trigonometric identity \(1/\cos^2 x = \sec^2 x\) we can simplify the result: $$\frac{1}{2} \sec^2 x + C$$ The final result for the integral is: $$\int \frac{\sin x}{\cos ^{3} x} d x = \frac{1}{2} \sec^2 x + C$$

Key Concepts

Integration by Substitution

Integration by substitution is a powerful technique in calculus. It's often used when an integral is difficult to solve directly. This method involves changing variables to make the integral more straightforward. By substituting part of the integral with a new variable, we can simplify complex expressions into basic forms.

In this problem, for example, we used the substitution \( u = \cos x \). The choice of \( u \) is crucial; it should be something whose derivative is also in the integral. This is why we picked \( \cos x \) because its derivative, \( - \sin x \), helped simplify the integral. After substituting and changing the variable of integration from \( x \) to \( u \), the integral becomes easier to handle, turning it into a power function that’s simple to integrate.

• Choose a substitution \( u \) that simplifies complex expressions.
• Derive \( du \) from \( u \) and use it to replace \( dx \).
• Rewrite the integral in terms of \( u \), then solve and substitute back.

Trigonometric Identities

Understanding trigonometric identities is key when working with integrals involving trigonometric functions. These identities allow you to simplify expressions and solve integrals that would otherwise be complex.

In the solution process for our exercise, we applied the identity \( 1/\cos^2 x = \sec^2 x \) to simplify the final expression. Trigonometric identities like this convert functions into either more straightforward or more useful forms, facilitating simplification.

For instance:

• \( \sin^2 x + \cos^2 x = 1 \)
• \( 1+\tan^2 x = \sec^2 x \)
• \( \sin 2x = 2 \sin x \cos x \)

These identities help you simplify integrals, manipulate expressions, and perform calculations easily. By knowing these identities, your problem-solving ability in calculus will greatly improve.

Definite and Indefinite Integrals

Integrals in calculus come in two forms: definite and indefinite. Each kind has its use and purpose.

Indefinite integrals focus on finding the antiderivative. They are generally expressed with a '+ C' to represent the constant of integration, since integrating a function removes any derivative information about constant terms. For example, in this exercise, we found the indefinite integral \( \int \frac{\sin x}{\cos^3 x} \, dx \) as \( \frac{1}{2} \sec^2 x + C \).

• Indefinite Integrals: Represent a family of functions.
• Do not have specific limits of integration.
• Include a constant 'C'.

Definite integrals, on the other hand, calculate the net area under a curve within certain bounds. They are represented with upper and lower limits of integration and do not include a '+ C'. They provide a numerical value that signifies, for instance, total accumulation or net change.

Calculus Problem-Solving Steps

Approaching calculus problems systematically is essential for success. Breaking down complex problems into manageable steps helps you see the bigger picture and find solutions.

In evaluating integrals, here’s a general path to follow:

• Analyze the integral to determine if substitution, parts, or another method works best.
• Choose and perform substitutions carefully to simplify the problem.
• Integrate the resulting expression using rules such as power rule, trigonometric identities, etc.
• After integration, remember to substitute back any variables you changed, reverting to the original variable.
• Simplify the final expression and check your result, often verifying through differentiation or other methods.

By following a structured approach, you ensure no detail is overlooked, making your problem-solving process efficient and effective. This methodical thinking is valuable not just in calculus but across mathematical disciplines.