Problem 13
Question
Use the Table of Integrals to compute each integral after manipulating the integrand in a suitable way. $$ \int \cos ^{2}(5 x-3) d x $$
Step-by-Step Solution
Verified Answer
The integral evaluates to \( \frac{1}{2}x + \frac{1}{20} \sin(10x - 6) + C \).
1Step 1: Recognize Trigonometric Identity
The integral you need to compute is \( \int \cos^2(5x - 3) \, dx \). To simplify this, we use a trigonometric identity: \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \). This allows us to express the integrand in a simpler form.
2Step 2: Apply the Trigonometric Identity
Using the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \), substitute \( \theta = 5x - 3 \). Consequently, \( \cos^2(5x - 3) = \frac{1 + \cos(2(5x - 3))}{2} \). This becomes \( \frac{1 + \cos(10x - 6)}{2} \).
3Step 3: Rewrite the Integral
Substitute the expression found in Step 2 back into the integral: \( \int \cos^2(5x - 3) \, dx = \int \frac{1 + \cos(10x - 6)}{2} \, dx \). Simplifying gives \( \frac{1}{2} \int (1 + \cos(10x - 6)) \, dx \).
4Step 4: Integrate Both Terms
To find \( \int \left( \frac{1}{2} \times 1 + \frac{1}{2} \cos(10x - 6) \right) \, dx \), split the integral into two parts: \( \frac{1}{2} \int 1 \, dx \) and \( \frac{1}{2} \int \cos(10x - 6) \, dx \).
5Step 5: Compute the First Integral
Compute \( \frac{1}{2} \int 1 \, dx \). This integral is straightforward, resulting in \( \frac{1}{2}x \).
6Step 6: Compute the Second Integral
For the second integral \( \frac{1}{2} \int \cos(10x - 6) \, dx \), use substitution. Let \( u = 10x - 6 \), then \( du = 10 \, dx \) or \( dx = \frac{du}{10} \). Substitute to get \( \frac{1}{2} \int \cos(u) \cdot \frac{du}{10} \), which simplifies to \( \frac{1}{20} \int \cos(u) \, du \).
7Step 7: Find the Integral of the Cosine Function
The integral \( \int \cos(u) \, du \) is \( \sin(u) + C \), hence \( \frac{1}{20} \int \cos(u) \, du = \frac{1}{20} \sin(u) + C \). Substitute back \( u = 10x - 6 \) to get \( \frac{1}{20} \sin(10x - 6) + C \).
8Step 8: Combine the Results
Combine the results from Step 5 and Step 7. The result of the original integral is \( \frac{1}{2}x + \frac{1}{20} \sin(10x - 6) + C \), where \( C \) is the constant of integration.
Key Concepts
Trigonometric IntegrationSubstitution MethodTrigonometric Identities
Trigonometric Integration
Trigonometric integration is a method used to integrate functions that involve trigonometric expressions. These functions often include terms like sine and cosine, which can be challenging to integrate directly.
To tackle these integrals, we utilize specific trigonometric identities and substitutions. These techniques simplify the function into a more manageable form. For example, if you encounter a function like \( \int \cos^2(\theta) \, d\theta \), you'd use the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \). Once simplified, the new integral form is generally easier to solve.
Understanding how to work with trigonometric integrals involves learning these identities and practicing substitutions. As you explore, pay attention to how these strategies simplify the integration process. When mastered, they reveal a seamless path to solve complex trigonometric expressions.
To tackle these integrals, we utilize specific trigonometric identities and substitutions. These techniques simplify the function into a more manageable form. For example, if you encounter a function like \( \int \cos^2(\theta) \, d\theta \), you'd use the identity \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \). Once simplified, the new integral form is generally easier to solve.
Understanding how to work with trigonometric integrals involves learning these identities and practicing substitutions. As you explore, pay attention to how these strategies simplify the integration process. When mastered, they reveal a seamless path to solve complex trigonometric expressions.
Substitution Method
The substitution method is a key technique in calculus used to simplify integrals. It involves changing the variable of integration to make an integral easier to evaluate. This technique is especially useful for integrals involving composite functions or specific trigonometric expressions.
In the given exercise, the substitution method was applied after the trigonometric identity. By substituting \( u = 10x - 6 \), and subsequently finding \( du = 10 \, dx \), the integral \( \int \cos(u) \, du \) becomes much simpler. The original integral transforms into the integral of a standard cosine function, which is straightforward to solve.
Here’s a quick recap of the process:
In the given exercise, the substitution method was applied after the trigonometric identity. By substituting \( u = 10x - 6 \), and subsequently finding \( du = 10 \, dx \), the integral \( \int \cos(u) \, du \) becomes much simpler. The original integral transforms into the integral of a standard cosine function, which is straightforward to solve.
Here’s a quick recap of the process:
- Identify a suitable substitution that will simplify part of the integrand.
- Replace the original variables with the substitution.
- Perform the integration with respect to the new variable.
- Substitute back to the original variable to express the solution.
Trigonometric Identities
Trigonometric identities are equations involving trigonometric functions that hold true for all values of the included variables. They are invaluable tools in calculus, particularly when managing integrals with trigonometric components.
One of the most frequently used identities in integration is \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \). This identity allows you to transform a square of a cosine function into a linear combination of cosine and constant terms, thus simplifying the integral.
These identities not only simplify integration tasks but also allow us to convert seemingly complex trigonometric expressions into simpler algebraic forms.
One of the most frequently used identities in integration is \( \cos^2(\theta) = \frac{1 + \cos(2\theta)}{2} \). This identity allows you to transform a square of a cosine function into a linear combination of cosine and constant terms, thus simplifying the integral.
These identities not only simplify integration tasks but also allow us to convert seemingly complex trigonometric expressions into simpler algebraic forms.
- Pythagorean Identities: Such as \( \sin^2(\theta) + \cos^2(\theta) = 1 \).
- Double Angle Identities: Used when dealing with expressions like \( \sin(2\theta) \) or \( \cos(2\theta) \).
- Sum-to-Product Identities: Useful for transforming sums into products.
Other exercises in this chapter
Problem 12
Use long division to write \(f(x)\) as a sum of a polynomial and a proper rational function. $$ f(x)=\frac{x^{3}+x}{x^{2}+x} $$
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In Problems 1-16, evaluate each indefinite integral by making the given substitution. $$ \int x e^{1-2 x^{2}} d x, \text { with } u=1-2 x^{2} $$
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Compute the Taylor polynomial of degree \(n\) about \(x=0\) for each function and compare the value of the function at the indicated point with the value of the
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Use the trapezoidal rule to approximate each integral with the specified value of \(n .\) Compare your approximation with the exact value. $$ \int_{1}^{3} x^{3}
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