Problem 74 - Applied Calculus

Question

Find the indefinite integrals. $$\int \sin (3 x) d x$$

Step-by-Step Solution

Answer

\(-\frac{1}{3} \cos(3x) + C\)

Step 1: Identify the formula for integration

Recall that the integral of \( \sin(ax) \) is given by the formula \( \int \sin(ax) \, dx = -\frac{1}{a} \cos(ax) + C \). Here, \( a = 3 \).

Step 2: Apply the integration formula

Using the formula from Step 1, replace \( a \) with 3: \( \int \sin(3x) \, dx = -\frac{1}{3} \cos(3x) + C \). This simplifies the integration using the constant multiplied to \( x \).

Step 3: Write the final answer

Conclude with the integrated function: \(-\frac{1}{3} \cos(3x) + C\). Make sure you include the integration constant \( C \), as this is an indefinite integral.

Key Concepts

Integral of sine function

When integrating trigonometric functions like sine, it's fundamental to understand the connection between different trigonometric identities and derivatives. The integral of the sine function is directly related to the derivative of the cosine function. This is because the derivative of \(-\cos(x)\) is \(\sin(x)\). Thus, when you see \(\sin(ax)\), you know its integral will involve \(\cos(ax)\) , but with a slight twist due to the chain rule. In this exercise, you're asked to integrate \(\sin(3x)\), which requires a bit more attention since there's a constant, \(3\), multiplying the variable.

• This constant affects the integral by changing its amplitude.
• As a result, the formula for the integral of \(\sin(ax)\) becomes \( -\frac{1}{a} \cos(ax)\) where \(a\) is the constant multiplier of \(x\).

Recognizing this pattern will help you integrate similar problems and appreciate the seamless transition between differentiation and integration.

Integration techniques

Integration is a powerful mathematical tool often likened to the reverse process of differentiation. In the realm of calculus, it's essential for determining areas under curves. When you face an integral like \(\int \sin(3x) \, dx\), it requires a simple application of a technique called the substitution or integration by recognition.- **Single Variable Substitution:** Sometimes, integrals come with constants attached to the variable, like \(3x\) in your problem. - To integrate such functions, acknowledge the constant, \(a\), and use \( -\frac{1}{a} \cos(ax)\).- **Pattern Recognition:** Another path is simply recognizing patterns from derivative formulas and flipping them back.Once you grasp these techniques, assignments like this become much more approachable. Begin by familiarizing yourself with basic integral formulas and recognize their patterns in more complex expressions.

Constant of integration

When dealing with indefinite integrals, such as \(\int \sin(3x) \, dx\), always remember the constant of integration, typically denoted as \(C\). This constant represents any constant value that could be added to the function being integrated. The reason for its necessity is that when differentiating a function, any constant term disappears. Thus, to compensate for this potential loss of information in indefinite integrals, we add \(C\). - **Why is \(C\) crucial?** - Every function with a constant added has the same derivative. - Indefinite integrals result from reversing differentiation, requiring \(C\) to account for all possible initial conditions or origins.- **Adding \(C\)** ensures the anti-derivative reflects any shifts along the \(y\)-axis.Recognizing the role of the constant in integration is vital. It ensures mathematical functions retain their universality, capturing all potential boundary conditions or specifics absent in pure differentiation.