Question

For the following exercises, find a general formula for the integrals. \(\int \sin a x \cos a x d x\)

Short answer

The general formula is \(-\frac{1}{4a} \cos(2ax) + C\).

Step-by-step solution

Identify a Trigonometric Identity

Notice that the integral \( \int \sin(ax) \cos(ax) \, dx \) involves a product of sine and cosine. We can use the trigonometric identity: \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \) to simplify the integrand. Let \( \theta = ax \), which implies \( \sin(ax) \cos(ax) = \frac{1}{2} \sin(2ax) \).

Substitute and Simplify the Integral

Substitute the identity into the integral: \(\int \sin(ax) \cos(ax) \, dx = \int \frac{1}{2} \sin(2ax) \, dx\). This simplifies to \( \frac{1}{2} \int \sin(2ax) \, dx \).

Integrate the Simplified Expression

Now, integrate \(\frac{1}{2} \int \sin(2ax) \, dx\). The integral of \( \sin(kx) \) is \( -\frac{1}{k} \cos(kx) + C \). Therefore, \(\frac{1}{2} \int \sin(2ax) \, dx = \frac{1}{2} \cdot -\frac{1}{2a} \cos(2ax) + C = -\frac{1}{4a} \cos(2ax) + C\).

Write the General Formula

The general formula for the integral \( \int \sin(ax) \cos(ax) \, dx \) is thus \( -\frac{1}{4a} \cos(2ax) + C \). Here, \( C \) represents the constant of integration.

Key concepts

Trigonometric Substitution

Trigonometric substitution is a powerful integration technique primarily used to simplify integrals involving trigonometric functions. When dealing with integrals like \( \int \sin(ax) \cos(ax) \, dx \), identifying a suitable trigonometric identity is key. The identity \( \sin(2\theta) = 2 \sin(\theta) \cos(\theta) \) is particularly useful here, allowing us to express the product of sine and cosine in a simpler form. By setting \( \theta = ax \), we transform the integral into \( \int \frac{1}{2} \sin(2ax) \, dx \). Using trigonometric identities in this way often reduces complex integrals into more manageable forms, facilitating easier computation further down the line.

Definite Integrals

Definite integrals are used to calculate the area under a curve from one point to another. Unlike indefinite integrals, which result in a general antiderivative with a constant \( C \), definite integrals compute a specific numerical value. Though the given exercise involves finding a general formula, it's crucial to understand that, when evaluating definite integrals, limits of integration are applied after finding the antiderivative. This provides an exact area or accumulation in the context of the problem.

Antiderivatives

Antiderivatives, also known as indefinite integrals, involve finding a function whose derivative results in the given integrand. In the exercise, the goal was to integrate \( \frac{1}{2} \sin(2ax) \), leading to its antiderivative. The integral of \( \sin(kx) \) is \( -\frac{1}{k} \cos(kx) + C \). Applying this formula, \( \int \sin(2ax) \, dx = -\frac{1}{2a} \cos(2ax) + C \), where \( C \) symbolizes the constant of integration. Understanding how to determine antiderivatives is essential in solving many integral problems.

Integration Techniques

Integration techniques encompass various methods to tackle integrals more effectively. One popular approach is recognizing patterns or identities that simplify the integrand, as seen with trigonometric substitution. Another technique is known as the u-substitution, but here the focus was on the trigonometric identity itself. For complex integrals involving products of trigonometric functions, leveraging such identities can drastically simplify calculations. Each technique provides a unique approach depending on the form of the integral, empowering you to solve otherwise challenging problems.