Question

Compute the following integrals using the guidelines for integrating powers of trigonometric functions. Use a CAS to check the solutions. (Note: Some of the problems may be done using techniques of integration learned previously.) \(\int \sin x \cos x d x\)

Short answer

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

Step-by-step solution

Identify the Integration Technique

The integral \( \int \sin x \cos x \, dx \) is in the product form of trigonometric functions. A suitable method for integrating this form is the use of the double angle identity for sine.

Apply the Double Angle Identity

Use the identity \( \sin 2x = 2 \sin x \cos x \) and rearrange it to match the integrand: \( \sin x \cos x = \frac{1}{2} \sin 2x \). This allows us to rewrite the integral as \( \int \sin x \cos x \, dx = \frac{1}{2} \int \sin 2x \, dx \).

Integrate Using Substitution

Now integrate \( \frac{1}{2} \int \sin 2x \, dx \). Let \( u = 2x \), then \( du = 2 \, dx \) or equivalently \( dx = \frac{1}{2} du \). Substitute into the integral: \( \frac{1}{2} \int \sin 2x \, dx = \frac{1}{4} \int \sin u \, du \).

Solve the Integral of Sine

The integral of \( \sin u \) is \( -\cos u + C \) where \( C \) is the constant of integration. Thus, \( \frac{1}{4} \int \sin u \, du = -\frac{1}{4} \cos u + C \).

Substitute Back to Original Variable

Replace \( u = 2x \) to convert back to the original variable: \( -\frac{1}{4} \cos 2x + C \). Now, we have the solution \( \int \sin x \cos x \, dx = -\frac{1}{4} \cos 2x + C \).

Verify the Solution with CAS

Use a CAS to enter the integral \( \int \sin x \cos x \, dx \). The software should confirm the antiderivative \( -\frac{1}{4} \cos 2x + C \).

Key concepts

Integration Techniques

Understanding integration techniques is crucial when solving integrals, especially those involving trigonometric functions. One popular method is substitution, where you change the variable to simplify the integral. In the provided exercise, the use of a trigonometric identity simplified the integral significantly. Once reduced, a substitution method was applied to further solve it.

Here's how it worked in this case:
* Identify a useful trigonometric identity.
* Substitute the new expression into the integral.
* Change variables if necessary, as shown with the change to "u".
The aim is always to transform the integral into an easier one to solve. Mastering these techniques is like having different tools in your toolbox, ready for any problem.

Trigonometric Identities

Trigonometric identities are key to simplifying complex integrals. They are equations involving trigonometric functions true for all values within their domains. In this exercise, the double angle identity, \[\sin 2x = 2 \sin x \cos x\], was used. It allowed the expression \( \sin x \cos x \) to be rewritten for easier integration.

These identities help by:
* Reducing the degrees of functions.
* Making substitution possible.
* Allowing for simpler computation.

Recognizing when and how to use a trigonometric identity is crucial. It transforms a seemingly complicated integral into something manageable. Practicing these can significantly improve your problem-solving skills in calculus.

Calculus Software Verification

Incorporating technology in solving calculus problems serves as an excellent means of verification. Calculus software, or CAS (Computer Algebra Systems), play an important role in confirming the accuracy of manually calculated integrals. After solving integrals step-by-step, input the original integral into a CAS tool.

Benefits of using a CAS:
* It provides quick verification of results.
* It checks complex calculations and reduces errors.
* It offers additional insights or methods not initially considered.

For students learning calculus, this practice supports and enhances traditional problem-solving skills. It also breeds confidence in their answers, giving assurance that the solutions derived are indeed correct.