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Chapter 8: Problem 36

Find the following indefinite integrals. $$\int 3 \sin 3 x \, d x$$

Short Answer

Expert verified
\(\int 3 \sin 3x \, dx = -\cos (3x) + C\)

Step by step solution

01

Identify the integral to be solved

The integral to be solved is \( \int 3 \sin (3x) \, dx \)
02

Determine the method to use

To solve this integral, use the formula for the integral of sine: \( \int \sin (ax) \, dx = -\frac{1}{a} \cos (ax) + C \). In this case, \(a = 3\).
03

Apply the formula

Substitute \( a = 3 \) into the formula: \( \int \sin (3x) \, dx = -\frac{1}{3} \cos (3x) + C \)
04

Consider the constant multiple

Since there is a constant \(3\) multiplied by the integral, it should be factored out: \( \int 3 \sin (3x) \, dx = 3 \left( -\frac{1}{3} \cos (3x) \right) + C \)
05

Simplify the expression

Simplify the integral: \( \int 3 \sin (3x) \, dx = - \cos (3x) + C \)

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Key Concepts

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

Integration by Substitution
The concept integration by substitution can often simplify more complex integrals by transforming them into a simpler form. The idea is to replace a part of the integral with a new variable, making the integral easier to solve.
For instance, in the integral \(\int 3 \,\sin(3x) \, dx\), we identify a substitution variable: \(u = 3x\).
Given that, we also write its derivative: \(du = 3 \, dx\).
With this substitution, the integral transforms as follows:
  • \( dx = du/3 \)
  • Our integral becomes \(\int 3\sin(u) \, du/3 \)
After simplifying, the integral reduces to \( \int \sin(u) \, du \).
This technique helps break down integrals that look challenging at first glance.
Integral of Sine Function
The integral of the sine function is critical in calculus.
Recall the basic integral formula: \( \int \sin(ax) \, dx = -\frac{1}{a}\cos(ax) + C \)
This standard result is key to solving trigonometric integrals by providing a straightforward way of obtaining antiderivatives.
In our example, \(a = 3\).
Plugging this value into the formula gives us: \( \int \sin(3x)\, dx = -\frac{1}{3} \cos(3x) + C \).
The negative sign indicates the direction of the function's antiderivative.
Keeping these standard results in mind makes solving integrals much quicker.
Constant Multiple Rule
The constant multiple rule in integration states that if you have a constant multiple inside an integral, you can factor it out.
Mathematically, this means: \( \int c \, f(x) \, dx = c \int f(x) \, dx \).
This rule greatly simplifies the integration process.
In our problem, the integral is: \(\int 3 \, \sin(3x) \, dx \).
Here, 3 is a constant outside the sine function.
According to the constant multiple rule:
  • Factor the constant out: \( 3 \int \sin(3x) \, dx \).
  • Find the antiderivative of \(\sin(3x)\): \(\int \sin(3x) \, dx = -\frac{1}{3} \cos(3x) + C\).
Multiply back the constant: \( 3 \left( - \frac{1}{3} \cos(3x) \right) + C \).
Simplify: \( -\cos(3x) + C \).
Applying the constant multiple rule helps streamline the calculations and reach the solution efficiently.

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