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Chapter 8: Problem 36

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

Short Answer

Expert verified
- (1/3) \cos(3x) + C

Step by step solution

01

- Identify the integrand

Examine the given integral \(\text{鈭� 3 sin(3x) dx}\). The integrand is \(3 \sin(3x)\).
02

- Use the constant multiple rule

One of the properties of integrals is that constants can be factored out. Apply this to the integral: \(\text{鈭� 3 sin(3x) dx = 3 鈭� sin(3x) dx}\).
03

- Apply the u-substitution method

Let \(u = 3x\). Then, differentiate \(u\) with respect to \(x\): \(du/dx = 3\) or \(du = 3 dx\). Solve for \(dx\): \(dx = du/3\). Rewrite the integral: \[\text{鈭� sin(u) (du/3) = (1/3) 鈭� sin(u) du} \]
04

- Integrate \(sin(u)\)

Recall the integral of \(sin(u)\): \(鈭� \sin(u) du = - \cos(u) + C\). Substitute back: \((1/3) 鈭� \sin(u) du = (1/3)(- \cos(u) + C)\).
05

- Substitute \(u = 3x\) back into the equation

Replace \(u \) with \(3x\) in the integrated function: \(= (1/3)(- \cos(3x) + C) \). Simplify the expression: \(- \cos(3x)/3 + C/3 = - (1/3) \cos(3x) + C\). Here, \(C/3\) is still an arbitrary constant, so it can be written as \(C\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

integrand
In calculus, the term 'integrand' refers to the function being integrated. For the given problem, \( \text{鈭� 3 sin(3x) dx} \), the integrand is \(3 \sin(3x)\). To solve the integral, we first need to correctly identify and understand the integrand. Recognizing the integrand allows us to apply different integration techniques appropriately. This function affects how we manipulate and simplify the integral during the solving process.
constant multiple rule
The constant multiple rule is a fundamental concept in integration. It states that any constant factor in the integrand can be taken out of the integral. This simplification makes the problem easier to solve. For instance, given \( \text{鈭� 3 sin(3x) dx} \), we use the constant multiple rule to rewrite the integral as \( 3 \text{鈭� sin(3x) dx} \). By separating the constant from the variable part of the integrand, we simplify the integral to focus on the variable function. This step essentially breaks the problem into more manageable parts.
u-substitution
U-substitution is a method used to solve more complex integrals by simplifying them into a basic form that we can integrate more easily. It involves substituting a part of the integrand with a new variable. For our integral, we let \(u = 3x \) and consequently \(du = 3 dx\). Solving for \(dx\), we get \(dx = du / 3\). We then substitute \(u\) and \(dx\) back into the integral: \( \text{鈭� 3 sin(3x) dx} = 3 \text{鈭� sin(u) (du / 3)} \). This simplifies to \( \text{鈭� sin(u) du} \). After integrating with respect to \(u\), we substitute \(3x\) back in for \(u\) to obtain the final result: \( -\frac{1}{3} \cos(3x) + C \).

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