Problem 37
Question
In Problems 1-40, find the general antiderivative of the given function. $$ f(x)=x^{-7}+3 x^{5}+\sin (2 x) $$
Step-by-Step Solution
Verified Answer
The general antiderivative is \( -\frac{1}{6}x^{-6} + \frac{1}{2}x^6 - \frac{1}{2}\cos(2x) + C \).
1Step 1: Identify the antiderivative of each term
To find the general antiderivative of a function, we take the antiderivative of each individual term separately. For the function \( f(x) = x^{-7} + 3x^5 + \sin(2x) \), we will find the antiderivative for each term: \( x^{-7} \), \( 3x^5 \), and \( \sin(2x) \).
2Step 2: Find the antiderivative of \( x^{-7} \)
The power rule for antiderivatives states that the antiderivative of \( x^n \) is \( \frac{x^{n+1}}{n+1} + C \) (where \( n eq -1 \)). For \( x^{-7} \), add 1 to the exponent to get \( x^{-6} \), and then divide by the new exponent: \( \frac{x^{-6}}{-6} \), which simplifies to \( -\frac{1}{6}x^{-6} \).
3Step 3: Find the antiderivative of \( 3x^5 \)
Again using the power rule, add 1 to the exponent 5 to get 6, and divide by the new exponent: \( \frac{3x^6}{6} \). Simplifying gives \( \frac{1}{2}x^6 \).
4Step 4: Find the antiderivative of \( \sin(2x) \)
The antiderivative of \( \sin(kx) \) is \( -\frac{1}{k}\cos(kx) + C \). For \( \sin(2x) \), the constant \( k \) is 2, so the antiderivative is \( -\frac{1}{2} \cos(2x) \).
5Step 5: Combine the results
Now, sum the antiderivatives of each term (and remember to include the constant of integration \( C \) for the general solution): \( -\frac{1}{6}x^{-6} + \frac{1}{2}x^6 - \frac{1}{2}\cos(2x) + C \).
Key Concepts
Power RuleIntegration TechniquesConstant of Integration
Power Rule
The power rule is a fundamental concept in calculus that simplifies the process of finding antiderivatives, which are essentially the reverse of derivatives. When you have a term like \( x^n \), and you want its antiderivative, the power rule is your go-to technique.
According to the power rule, the antiderivative of \( x^n \) is given by \( \frac{x^{n+1}}{n+1} \) plus a constant \( C \). This rule doesn't apply when \( n = -1 \) because that would involve division by zero, and instead, for \( x^{-1} \), the antiderivative is \( \ln |x| \).
For our example function \( f(x) = x^{-7} + 3x^5 + \sin(2x) \), the power rule is applied to the terms \( x^{-7} \) and \( 3x^5 \). You add 1 to the exponent, creating new terms \( x^{-6} \) and \( x^6 \), and then divide by the new exponent.
According to the power rule, the antiderivative of \( x^n \) is given by \( \frac{x^{n+1}}{n+1} \) plus a constant \( C \). This rule doesn't apply when \( n = -1 \) because that would involve division by zero, and instead, for \( x^{-1} \), the antiderivative is \( \ln |x| \).
For our example function \( f(x) = x^{-7} + 3x^5 + \sin(2x) \), the power rule is applied to the terms \( x^{-7} \) and \( 3x^5 \). You add 1 to the exponent, creating new terms \( x^{-6} \) and \( x^6 \), and then divide by the new exponent.
- The antiderivative of \( x^{-7} \) becomes \( -\frac{1}{6}x^{-6} \).
- The antiderivative of \( 3x^5 \) becomes \( \frac{1}{2}x^6 \).
Integration Techniques
Integration techniques are essential strategies in calculus used to find the antiderivatives of more complicated functions. Apart from the basic power rule, other techniques include substitution, parts, and dealing with trigonometric functions.
In our problem, we encounter the term \( \sin(2x) \), which incorporates a trigonometric function. The general method to find the antiderivative of \( \sin(kx) \) involves the formula \( -\frac{1}{k}\cos(kx) + C \). This technique is a result of recognizing how differentiation and integration handle the 'constant' k. For \( \sin(2x) \), \( k \) is 2, so the resulting antiderivative is \( -\frac{1}{2}\cos(2x) \).
Using the correct technique for the function type simplifies the integration process and helps in accurately finding the antiderivative.
In our problem, we encounter the term \( \sin(2x) \), which incorporates a trigonometric function. The general method to find the antiderivative of \( \sin(kx) \) involves the formula \( -\frac{1}{k}\cos(kx) + C \). This technique is a result of recognizing how differentiation and integration handle the 'constant' k. For \( \sin(2x) \), \( k \) is 2, so the resulting antiderivative is \( -\frac{1}{2}\cos(2x) \).
Using the correct technique for the function type simplifies the integration process and helps in accurately finding the antiderivative.
Constant of Integration
The constant of integration, often denoted as \( C \), plays a crucial role in calculus, especially when determining antiderivatives or indefinite integrals.
It represents an unknown constant because when differentiating, the derivative of a constant is zero, meaning it disappears from calculations. When we find antiderivatives, we must include \( C \) to account for any possible constant that was present before differentiation.
It represents an unknown constant because when differentiating, the derivative of a constant is zero, meaning it disappears from calculations. When we find antiderivatives, we must include \( C \) to account for any possible constant that was present before differentiation.
- For the antiderivative \( -\frac{1}{6}x^{-6} + \frac{1}{2}x^6 - \frac{1}{2}\cos(2x) \), we add \( C \) to ensure we cover all potential original functions.
- This general solution acknowledges that there are infinitely many specific solutions, each differing by only a constant number.
Other exercises in this chapter
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