Problem 66 - Calculus Single Variable

Question

When \(\sin (x)\) and \(\cos (x)\) are both raised to the same positive power in an integrand, the identity \(\sin (2 x)=2 \sin (x) \cos (x)\) may be used to simplify the integral. Use this observation as the basis for calculation of the integrals. $$ \int \cos (x) \sin (x) d x $$

Step-by-Step Solution

Answer

The integral is \( -\frac{1}{4} \cos(2x) + C \).

Step 1: Identify the integrand

We are given the integral \( \int \cos(x) \sin(x) \, dx \). The integrand is a product of sine and cosine, both to the first power.

Step 2: Use the double angle identity

Recall the double angle identity: \( \sin(2x) = 2 \sin(x) \cos(x) \). To fit our integrand, we can rewrite it in terms of this identity. Notice that \( \cos(x) \sin(x) = \frac{1}{2} \sin(2x) \).

Step 3: Substitute into the integral

Replace \( \cos(x) \sin(x) \) in the integrand with \( \frac{1}{2} \sin(2x) \). The integral now becomes \( \int \frac{1}{2} \sin(2x) \, dx \).

Step 4: Simplify and integrate

Factor out the constant \( \frac{1}{2} \) from the integral for simplicity. The integral now is \( \frac{1}{2} \int \sin(2x) \, dx \). Use the basic integration rule for \( \sin(kx) \), which is \( \int \sin(kx) \, dx = -\frac{1}{k} \cos(kx) + C \).

Step 5: Compute the integral

Using the integration rule from Step 4 with \( k = 2 \), gives \( \int \sin(2x) \, dx = -\frac{1}{2} \cos(2x) + C \). Thus, \( \frac{1}{2} \int \sin(2x) \, dx = \frac{1}{2}(-\frac{1}{2} \cos(2x)) + C = -\frac{1}{4} \cos(2x) + C \).

Step 6: State the final answer

The integral of \( \cos(x) \sin(x) \, dx \) is \( -\frac{1}{4} \cos(2x) + C \), where \( C \) is the constant of integration.

Key Concepts

Integration Techniques

Integration techniques are essential tools in calculus for finding antiderivatives or integral expressions. When dealing with the product of trigonometric functions like \( \cos(x) \sin(x) \), recognizing useful identities can simplify the process.
One effective strategy is to use trigonometric identities to transform difficult integrals into simpler ones. For example, in the problem with \( \int \cos(x) \sin(x) \, dx \), using the double angle identity is an integral technique that simplifies computation.

• First, identify the integrand's structure. In this case, both sine and cosine are to the first power.
• Use the identity \( \sin(2x) = 2\sin(x)\cos(x) \) to express the integrand in a simpler form. Here, \( \cos(x) \sin(x) \) can be rewritten as \( \frac{1}{2} \sin(2x) \).

By substituting back into the original integral, the problem becomes a standard form that is easier to solve. This approach highlights the power of integration techniques in calculus, emphasizing that recognizing underlying patterns is key to problem-solving.

Trigonometric Identities

Trigonometric identities are expressions involving trigonometric functions that are always true for any angle. They are crucial for solving integrals involving trigonometric functions.

The identity used in this problem is the double angle identity: \( \sin(2x) = 2\sin(x)\cos(x) \). It allows us to simplify the integrand \( \cos(x) \sin(x) \) by expressing it in terms of a single sine function at double the angle.

• This transformation converts a product of functions into a single function, making the integral easier to evaluate.

The use of trigonometric identities transforms challenging integrals and simplifies calculations by providing alternative expressions. When tackling similar integrals, always recall basic trigonometric transformations to simplify your work.

Integral Calculus

Integral calculus is the study of integrals and their properties. It focuses not only on finding the antiderivatives but also understanding the area under curves represented by functions.

In this exercise, we examined the integral \( \int \cos(x)\sin(x)\,dx \), which seems complex at first. However, by using the right techniques and identities, it becomes straightforward.

The solution involves:

• Substituting the integrand with \( \frac{1}{2} \sin(2x) \) due to the trigonometric identity \( \sin(2x) = 2\sin(x)\cos(x) \).
• Recognizing the simpler form \( \frac{1}{2} \int \sin(2x)\, dx \), which is solved using standard integration rules.

Integral calculus combines analytical skills with creativity. Solving such integrals requires understanding not just techniques but also how functions and their properties interrelate.