Problem 19
Question
Find the general antiderivative. $$\int(3 \cos x-1 / x) d x$$
Step-by-Step Solution
Verified Answer
The general antiderivative of the given function is \(3 \sin x - \ln |x| + C\).
1Step 1: Identify the functions to be integrated
We need to integrate \(3 \cos x\) and \(-1 / x\). It's important to note that the integral of a sum is the sum of integrals. Thus, the integral of \(3 \cos x - 1 / x\) is the integral of \(3 cos x\) minus the integral of \(1 / x\).
2Step 2: Find the antiderivative of \(3 \cos x\)
The antiderivative of \(\cos x\) is \(\sin x\). The constant 3 is factored out of the integral, so the integral of \(3 \cos x\) is \(3 \sin x\).
3Step 3: Find the antiderivative of \(1 / x\)
The antiderivative of \(1 / x\) is \(\ln |x|\). Because the original function was subtracting this term, it becomes minus in the antiderivative. So, the integral of \(-1 / x\) is \(-\ln |x|\).
4Step 4: Combine the antiderivatives and add the constant of integration
The general antiderivative of the given function is the sum of the antiderivatives from Step 2 and Step 3 plus a constant. This constant arises due to the indefinite integral, and represents all the possible constants that were differentiated to give the original function. Therefore the final solution is \(3 \sin x - \ln |x| + C\).
Key Concepts
Integration TechniquesIndefinite IntegralTrigonometric FunctionsLogarithmic Functions
Integration Techniques
Integration techniques play a crucial role in finding antiderivatives, which are essential in calculus. The process of integration reverses differentiation and provides insight into the accumulated quantity over an interval. In our exercise, we encounter the integration of a sum, which involves two distinct functions: a trigonometric function and a logarithmic function.
When dealing with a sum of functions, a key technique is to recognize that the integral of a sum equals the sum of the integrals. This allows us to tackle each function individually, simplifying the problem.
When dealing with a sum of functions, a key technique is to recognize that the integral of a sum equals the sum of the integrals. This allows us to tackle each function individually, simplifying the problem.
- Factor out constants: When a function is multiplied by a constant, factor it out to simplify calculations.
- Integration by parts: While not needed here, it's helpful in complex integration situations.
- Substitution: Useful for integrating composite functions, though not required in this example.
Indefinite Integral
An indefinite integral, unlike a definite integral, doesn't have upper and lower limits. Instead, it represents a family of functions whose derivative corresponds to the integrand. In essence, it finds the general antiderivative.
In our problem, we determine the indefinite integral of the function given. We add a constant of integration, usually denoted as 'C', because the derivative of any constant is zero. This constant accounts for all possible vertical shifts of the antiderivative function.
In our problem, we determine the indefinite integral of the function given. We add a constant of integration, usually denoted as 'C', because the derivative of any constant is zero. This constant accounts for all possible vertical shifts of the antiderivative function.
- The result is expressed as an equation without specific bounds.
- The addition of 'C' makes it a general solution, representing any constant added to the function without altering its derived state.
Trigonometric Functions
Trigonometric functions appear frequently in calculus problems and have well-defined antiderivatives. In our exercise, we deal with the function \(3 \cos x\). The process involves recognizing known antiderivatives of basic trigonometric functions.
The antiderivative of \(\cos x\) is \(\sin x\), which is a fundamental identity in calculus. Because our function is \(3 \cos x\), we multiply the antiderivative by the constant 3 to get \(3 \sin x\).
The antiderivative of \(\cos x\) is \(\sin x\), which is a fundamental identity in calculus. Because our function is \(3 \cos x\), we multiply the antiderivative by the constant 3 to get \(3 \sin x\).
- A valuable tip is to memorize the derivatives and antiderivatives of fundamental trigonometric functions, as they frequently recur.
- These functions help describe periodic behavior and oscillations.
Logarithmic Functions
Logarithmic functions, such as \(1/x\), often require careful handling during integration. The antiderivative of \(1/x\) is the natural logarithm, \(\ln|x|\). The absolute value ensures the logarithm remains valid for any nonzero x.
In our integration of \(-1/x\), the antiderivative is \(-\ln|x|\), as the negative sign directly affects both the function and its integral.
In our integration of \(-1/x\), the antiderivative is \(-\ln|x|\), as the negative sign directly affects both the function and its integral.
- Remember that the integral of \(1/x\) leads to a logarithm, making it a unique case in calculus.
- Logarithmic functions help model growth processes, like population growth or decay.
- Pay attention to domain restrictions; logarithms are only defined for positive arguments.
Other exercises in this chapter
Problem 19
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Compute the sum and the limit of the sum as \(n \rightarrow \infty.\) $$\sum_{i=1}^{n} \frac{1}{n}\left[\left(\frac{i}{n}\right)^{2}+2\left(\frac{i}{n}\right)\r
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Evaluate the indicated integral. $$\int \tan 2 x \, d x$$
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