Problem 1

Question

Evaluate the indefinite integrals in Exercises $1-12$ by using the given substitutions to reduce the integrals to standard form. $$ \int \sin 3 x d x, \quad u=3 x $$

Step-by-Step Solution

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Answer

The indefinite integral is $-\frac{1}{3}\cos(3x) + C$.

1Step 1: Identify the substitution

The problem provides a substitution: let $ u = 3x $. This substitution will make the integrand easier to handle by changing the variable.

2Step 2: Calculate the differential

We need to find the differential of $ u $. Since $ u = 3x $, differentiate both sides with respect to $ x $ to get $ du = 3 \, dx $.

3Step 3: Express $ dx $ in terms of $ du $

From the equation $ du = 3 \, dx $, solve for $ dx $ to find $ dx = \frac{du}{3} $.

4Step 4: Substitute into the integral

Substitute $ u = 3x $ and $ dx = \frac{du}{3} $ into the integral to get $ \int \sin(u) \cdot \frac{du}{3} $. This simplifies the integral to $ \frac{1}{3} \int \sin(u) \, du $.

5Step 5: Integrate in terms of $ u $

The integral of $ \sin(u) $ is $ -\cos(u) + C $. Thus, the integral becomes $ \frac{1}{3}(-\cos(u)) + C = -\frac{1}{3}\cos(u) + C $.

6Step 6: Substitute back in terms of $ x $

Replace $ u $ with $ 3x $ to get the final solution. The indefinite integral is $ -\frac{1}{3}\cos(3x) + C $.

Key Concepts

Integration TechniquesSubstitution MethodTrigonometric Integrals

Integration Techniques

Understanding integration techniques is crucial in mastering calculus, as they give us the tools to find values for indefinite integrals. Indefinite integrals are expressions that represent the family of all antiderivatives of a function, effectively the reverse operation of differentiation. When solving indefinite integrals, we want to find a function whose derivative is the provided function. There are various integration techniques that help to tackle different types of integrals, including:

Basic Integration: Applying basic rules, such as the power rule, directly to find the integral.
Substitution Method: Simplifying integrals by changing variables to a more manageable form.
Integration by Parts: Useful when dealing with products of functions.
Partial Fractions: Decomposing rational functions into simpler fractions for ease of integration.

For the given problem, substituting a new variable simplifies the integration process significantly. Integrating requires carefully choosing the right technique based on the function type, ensuring that we can reduce it to a standard integrable form.

Substitution Method

The substitution method, often known as "u-substitution," is an essential technique for integrating more complex functions. It involves changing the variable of integration to simplify the function into a more standard form that is easier to integrate. This method takes advantage of the chain rule in reverse.The steps to using the substitution method typically include:

Choosing a new variable, such as $ u $, which is typically a function within the original integrand.
Calculating the differential $ du $ in terms of $ dx $, which involves differentiating $ u $.
Substituting both $ u $ and $ dx $ in terms of $ du $ to rewrite the integral.
Integrating the simplified function in terms of $ u $.
Replacing $ u $ with the original variable to express the integral back in terms of $ x $.

In our exercise, the original integral $ \int \sin 3x \, dx $ is transformed using $ u = 3x $ and $ dx = \frac{du}{3} $. This reduces the problem to the familiar form $ \frac{1}{3} \int \sin u \, du $, making it straightforward to integrate.

Trigonometric Integrals

Trigonometric integrals involve integrating functions that contain trigonometric functions like sine, cosine, and tangent. These integrals often appear in calculus and are simplified using various techniques depending on their complexity and form.For simple trigonometric integrals, standard formulas can be directly applied:

The integral of $ \sin x $ is $ -\cos x + C $.
The integral of $ \cos x $ is $ \sin x + C $.
The integral of $ \tan x $ can be expressed using $ -\ln|\cos x| + C $.

In our exercise, we deal with $ \int \sin 3x \, dx $, which involves multiple steps to simplify the trigonometric function using substitution. By choosing $ u = 3x $, we transform the integral into one involving $ \sin u $, allowing us to apply the standard formula for the sine function. After substitution, it becomes straightforward to integrate into $ -\frac{1}{3} \cos(3x) + C $, demonstrating how substitution helps in managing trigonometric expressions.

Problem 1

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