Problem 17 - Calculus: Graphical, Numerical, Algebraic

Question

In Exercises \(17-24,\) use the indicated substitution to evaluate the integral. Confirm your answer by differentiation. $$\int \sin 3 x d x, \quad u=3 x$$

Step-by-Step Solution

Answer

The integral \(\int \sin 3x dx\) evaluates to \(-\cos(3x)\), confirmed by differentiation.

Step 1: Apply the substitution

Replace \(x\) with \( \frac{u}{3} \) in the integral. Therefore, your integral becomes \(\int \sin u du\). Do this because when \(u = 3x\), then \(du = 3 dx\), which means that \(dx = du/3\).

Step 2: Evaluate the integral

Now, you can proceed to integrate \(\int \sin(u) du\). The integral of \(\sin(u)\) with respect to \(u\) is \(-\cos(u)\). Hence, the integral becomes \(-\cos(u)\).

Step 3: Replace u with the original variable

Make sure to replace \(u\) back to \(3x\) which was our original variable. Thus, our solution becomes \(-\cos(3x)\).

Step 4: Confirm your answer by differentiation

Differentiate the result from step 3 which is \(-\cos(3x)\) with respect to \(x\) to confirm the solution. The derivative of \(-\cos(3x)\) using the chain rule is \(\sin(3x) * 3 = 3 \sin(3x)\), which is the original integrand, confirming that the solution is correct.

Key Concepts

Trigonometric Integration

Trigonometric integration refers to the process of finding the antiderivative, or integral, of trigonometric functions. This is a fundamental skill in calculus, as trigonometric functions are prevalent in many areas of mathematics and physics. There are various techniques to integrate these functions, but some common integrals include:

• For \( \sin(x) \), the integral is \( -\cos(x) + C \), where \(C\) is the constant of integration.
• For \( \cos(x) \), the integral is \( \sin(x) + C \).
• For \( \tan(x) \), the integral is \( -\ln|\cos(x)| + C \).
• For \( \sec^2(x) \), the integral is \( \tan(x) + C \).
• For \( \sec(x)\tan(x) \), the integral is \( \sec(x) + C \).
• For \( \csc(x)\cot(x) \), the integral is \( -\csc(x) + C \).

To execute trigonometric integration, it's often necessary to use identities such as \( \sin^2(x) + \cos^2(x) = 1\) or half-angle and double-angle formulas to simplify the integrand before integrating.

Substitution Method in Integration

The substitution method in integration, also called the 'u-substitution', is a technique that simplifies the integration process by transforming the integral into a simpler form that is easier to evaluate. The basic idea is to choose a substitution that converts the integral into a basic form whose antiderivative is known.

The process involves identifying a part of the integrand that can be substituted with a variable \(u\), differentiating \(u\) to find \(du\), and rewriting the integral in terms of \(u\) and \(du\). This often simplifies the integral, making it straightforward to find the antiderivative. After integrating with respect to \(u\), you then substitute back to the original variable to find the integral in terms of the original problem's variable.

Integration by Substitution

Integration by substitution is essentially an application of the chain rule of differential calculus in reverse. When we have an integral that cannot be easily integrated in its current form, we look for a substitution that makes it easier. The substitution is usually chosen so that the differential \(du\) corresponds to a part of the integral that can be canceled out or simplified.

For example, when integrating a function like \( \sin(3x) \), we can let \(u = 3x\). Then, we find \(du/dx = 3\), which implies \(du = 3dx\), and our integral in terms of \(x\) becomes much more manageable in terms of \(u\), transforming into \( \int \sin(u) \, du\). After integrating with respect to \(u\), the final step is to replace \(u\) with \(3x\) to return to the variable in the original question.

Confirming Integration by Differentiation

Confirming integration by differentiation is a technique that provides reassurance that we have correctly found the antiderivative of a function. Once we have the solution to an indefinite integral, we differentiate it with respect to the original variable.

If the differential of our result is the integrand we started with, we can be confident that our integration was carried out correctly. This is because differentiation is the reverse operation to integration.

In our example, after integrating \( \sin(3x) \), we obtained \( -\cos(3x) \). To confirm this result, we differentiate it to get \( 3 \sin(3x) \), which, aside from the constant factor of \(3\), matches the original integrand, verifying our solution. It's important to remember to account for the chain rule when differentiating composite functions, as seen in this process.