Problem 23 - AP Calculus with 8 Practice Tests 15th

Question

\(\int \sin \theta \cos \theta d \theta=\) (A) \(-\frac{\sin ^{2} \theta}{2}+C\) (B) \(-\frac{1}{4} \cos 2 \theta+C\) (C) \(\frac{\cos ^{2} \theta}{2}+C\) (D) \(\frac{1}{2} \sin 2 \theta+C\)

Step-by-Step Solution

Answer

The correct answer is (B): \(-\frac{1}{4} \cos 2\theta + C\).

Step 1: Use Trigonometric Identity

To solve the integral \( \int \sin \theta \cos \theta \, d\theta \), we can use the trigonometric identity \( \sin 2\theta = 2 \sin \theta \cos \theta \). This allows us to rewrite the integrand in terms of \( \sin 2\theta \).

Step 2: Substitute Using the Identity

Using the identity from Step 1, we can express \( \sin \theta \cos \theta \) as \( \frac{1}{2} \sin 2\theta \). Therefore, the integral becomes: \[ \int \sin \theta \cos \theta \, d \theta = \int \frac{1}{2} \sin 2\theta \, d\theta. \]

Step 3: Integrate

Integrate \( \frac{1}{2} \sin 2\theta \) with respect to \( \theta \). The integral of \( \sin 2\theta \) is \( -\frac{1}{2} \cos 2\theta \). Therefore, the integral becomes:\[ \int \frac{1}{2} \sin 2\theta \, d\theta = \frac{1}{2} \left(-\frac{1}{2} \cos 2\theta \right) + C = -\frac{1}{4} \cos 2\theta + C. \]

Step 4: Select the Correct Option

Compare the result from Step 3 with the given options. The answer is option (B): \(-\frac{1}{4} \cos 2\theta + C\).

Key Concepts

Trigonometric Identities

Understanding trigonometric identities is crucial for solving many calculus problems, especially those that involve integration. Trigonometric identities are equations that are true for all values of the variables involved where both sides of the equation are defined. Certain identities simplify complex trigonometric expressions, making them easier to work with.
For instance, the double-angle identity for sine, \( \sin 2\theta = 2 \sin \theta \cos \theta \), provides a way to simplify integrals involving sine and cosine products. This identity directly helps transform an integral like \( \int \sin \theta \cos \theta \, d\theta \) into a simpler form.
Utilizing these identities is like having a toolkit; it's all about knowing which tools (or identities) to use and when to use them. These tools help in rewriting expressions in a more manageable way, often as a single trigonometric function or by reducing the power of trigonometric terms.

Substitution Method

The substitution method is a powerful technique to make integrals more workable by changing the variable of integration to something simpler. Think of substitution as a method to "re-label" parts of an integral. It often goes hand in hand with trigonometric identities, as seen in our original solution.
When using the double-angle identity, we substituted \( \sin \theta \cos \theta \) with \( \frac{1}{2} \sin 2\theta \). The integral became \( \int \frac{1}{2} \sin 2\theta \, d\theta \). Such substitution is critical when there's a direct, more manageable equivalent for a trickier expression.
In general, when you face complex functions that are hard to integrate directly, looking for substitution opportunities by identifying patterns related to known derivatives or identities can simplify your task greatly.

Definite Integrals

Definite integrals provide a means to calculate the area under a curve between two points, and they yield a numerical answer rather than an expression containing a variable. However, in the context of our exercise, the focus was on integrating a trigonometric function to find an indefinite integral, but learning about definite integrals is the next logical step.
While definite integrals involve evaluating an integral from a specific lower to an upper bound, indefinite integrals, like the one given here, do not have such bounds and instead result in a family of functions, represented by a constant \( C \).
To move from an indefinite to a definite integral, integrate the function and evaluate the difference between the antiderivative at the upper limit and the lower limit. This process is summarized by the Fundamental Theorem of Calculus. With practice, you'll get comfortable switching contexts, depending on whether a solution or a numeric/resulting integral value is required.