Question

Evaluate each of the following integrals by u-substitution. \(\int \sin ^{7}(2 x) \cos (2 x) d x\)

Short answer

The evaluated integral is \( \frac{(\sin(2x))^8}{16} + C \).

Step-by-step solution

Choose a Substitution

We want to simplify the integral by making a substitution. Notice the presence of both \( \sin(2x) \) and its derivative \( \cos(2x) \). This suggests the substitution \( u = \sin(2x) \).

Differentiate the Substitution

Differentiate \( u = \sin(2x) \) with respect to \( x \): \[\frac{du}{dx} = 2\cos(2x)\]This implies \( du = 2\cos(2x)\, dx \) or \( \frac{1}{2}du = \cos(2x)\, dx \).

Substitute and Simplify the Integral

Substitute \( u = \sin(2x) \) and \( \frac{1}{2}du = \cos(2x)\, dx \) into the integral:\[\int \sin^7(2x) \cos(2x)\, dx = \int u^7 \cdot \frac{1}{2} du = \frac{1}{2} \int u^7 du\]

Integrate with respect to u

Now integrate \( \frac{1}{2} \int u^7 du \):\[= \frac{1}{2} \cdot \frac{u^8}{8} = \frac{u^8}{16}\]

Substitute Back to x

Re-substitute \( u = \sin(2x) \) back into the expression:\[\frac{u^8}{16} = \frac{(\sin(2x))^8}{16}\]

Add the Constant of Integration

Since the problem involves an indefinite integral, add the constant of integration \( C \):\[\frac{(\sin(2x))^8}{16} + C\]

Key concepts

Integration Techniques

Integration techniques are powerful tools in calculus that help us find antiderivatives, or evaluate integrals, effectively. Among these, **u-substitution** is a common method used to simplify integrals, particularly when the integrand is a composite function. The idea behind u-substitution is to replace a part of the integral with a single variable, usually called "u," which simplifies the integration process by transforming the variable of integration.

Here’s how it commonly works:

• Identify a part of the integrand that, if substituted, will simplify the integral with its derivative also appearing in the expression. This step often involves checking if the derivative of a chosen function is present in the remaining terms.
• Convert the differential in terms of the new variable "u" to accompany the substitution.
• Rewrite the integral in terms of "u" and integrate.
• Substitute back the original expressions in terms of "x" after integration to find the final solution.

In the given exercise, we used u-substitution where we set \( u = \sin(2x) \). Recognizing that its derivative \( \cos(2x) \) was also part of the integrand allowed us to transform and integrate the simpler form in terms of 'u'.

Indefinite Integrals

Indefinite integrals are a fundamental concept of calculus, representing a family of functions that describe antiderivatives of a function. While definite integrals compute the area under the curve to yield a fixed number, indefinite integrals express a general solution that includes all possible antiderivatives.The solution to an indefinite integral is always accompanied by a constant of integration, denoted as \( C \). This constant accounts for any vertical shifts of the antiderivative curve, since differentiation removes such constant terms.When performing u-substitution, as in our exercise, it’s important to remember:

• The integral result in terms of "u" is still an indefinite integral, and it includes this constant of integration.
• After substituting back to "x," the constant \( C \) should still be included in the answer.

For our example, after determining \( \frac{u^8}{16} \), the final expression was updated to \( \frac{(\sin(2x))^8}{16} + C \) to respect the requirement for this constant.

Trigonometric Integrals

Trigonometric integrals often involve integrals where the functions \( \sin x \), \( \cos x \), and their powers are key components. These kinds of integrals commonly require specific techniques like trigonometric identities or substitution methods, such as u-substitution, to simplify and solve them efficiently.In our specific exercise, the integral \( \int \sin^7(2x) \cos(2x) \, dx \) is tackled using u-substitution:

• Set \( u = \sin(2x) \) to simplify the powers of trigonometric functions. This substitution helped transform the problem into a simpler polynomial integral of \( \int u^7 \, du \).
• Recognize trigonometric derivatives where \( \cos(2x) \, dx \) directly aligns with the differentiation of \( \sin(2x) \), ensuring a smooth substitution process.
• Convert back to the original variable after integrating, leveraging the inverse substitution of the initial trigonometric substitution.

Understanding these techniques for evaluating trigonometric integrals is crucial for solving more complex integrations involving trigonometric functions.