Problem 30
Question
Find the general antiderivative of the given function. $$ f(x)=-3 \sin \left(\frac{\pi}{3} x\right)+4 \cos \left(-\frac{\pi}{4} x\right) $$
Step-by-Step Solution
Verified Answer
The general antiderivative is \( \frac{9}{\pi} \cos \left(\frac{\pi}{3} x\right) + \frac{16}{\pi} \sin \left(\frac{\pi}{4} x\right) + C \).
1Step 1: Analyze the Function
The function is given as \( f(x) = -3 \sin \left(\frac{\pi}{3} x\right) + 4 \cos \left(-\frac{\pi}{4} x\right) \). We need to find the antiderivative of each term separately. The structure suggests using basic trigonometric antiderivative formulas.
2Step 2: Find Antiderivative of First Term
The first term of the function is \(-3 \sin \left(\frac{\pi}{3} x\right)\). The antiderivative of \( \sin(ax) \) is \(-\frac{1}{a} \cos(ax)\). Using \( a = \frac{\pi}{3} \), we find that the antiderivative of this term is \( 3 \times \frac{1}{\frac{\pi}{3}} \cos \left(\frac{\pi}{3} x\right) = \frac{9}{\pi} \cos \left(\frac{\pi}{3} x\right) \).
3Step 3: Find Antiderivative of Second Term
The second term is \( 4 \cos \left(-\frac{\pi}{4} x\right) \). Use the formula; the antiderivative of \( \cos(ax) \) is \( \frac{1}{a} \sin(ax) \). Here, \( a = -\frac{\pi}{4} \). Therefore, the antiderivative is \( 4 \times -\frac{1}{\frac{\pi}{4}} \sin \left(-\frac{\pi}{4} x\right) = -\frac{16}{\pi} \sin \left(-\frac{\pi}{4} x\right) \). Use \( \sin(-x) = -\sin(x) \) to rewrite it as \( \frac{16}{\pi} \sin \left(\frac{\pi}{4} x\right) \).
4Step 4: Combine the Antiderivatives with the Constant of Integration
Combine the antiderivatives of both terms: \[ F(x) = \frac{9}{\pi} \cos \left(\frac{\pi}{3} x\right) + \frac{16}{\pi} \sin \left(\frac{\pi}{4} x\right) + C \] where \( C \) is the constant of integration. This represents the general antiderivative of the function \( f(x) \).
Key Concepts
Trigonometric FunctionsIntegration TechniquesConstant of Integration
Trigonometric Functions
Trigonometric functions like sine and cosine often appear in calculus, especially when finding antiderivatives. In the context of the exercise, we deal with the trigonometric functions \( \sin(x) \) and \( \cos(x) \). These functions describe the relationships of angles and sides in right-angled triangles.
However, they also appear frequently in mathematical equations due to their periodic nature. Understanding general properties of these functions helps in calculus:
However, they also appear frequently in mathematical equations due to their periodic nature. Understanding general properties of these functions helps in calculus:
- Sine Function (\( \sin \)): This function gives the ratio of the length of the opposite side to the hypotenuse in a right triangle. Its antiderivative helps us find the original function before differentiation by reversing the sine's effect.
- Cosine Function (\( \cos \)): Cosine provides the ratio of the adjacent side to the hypotenuse. It's another essential trigonometric function with straightforward antiderivative calculations.
Integration Techniques
Integration techniques help us reverse the effect of calculus operations such as differentiation. A common goal is to find a function given its derivative, known as an antiderivative.
For trigonometric functions, antiderivatives can be found using specific integration formulas. Each trigonometric function typically corresponds to a simple integration formula:
In the exercise, we break down the function into simpler parts that fit standard formulas, which makes it easier to apply the integration rules. By handling each segment of the composite function separately, integration becomes more manageable and less error-prone.
For trigonometric functions, antiderivatives can be found using specific integration formulas. Each trigonometric function typically corresponds to a simple integration formula:
- The integral of \( \sin(ax) \) is \( -\frac{1}{a} \cos(ax) \).
- The integral of \( \cos(ax) \) is \( \frac{1}{a} \sin(ax) \).
In the exercise, we break down the function into simpler parts that fit standard formulas, which makes it easier to apply the integration rules. By handling each segment of the composite function separately, integration becomes more manageable and less error-prone.
Constant of Integration
The constant of integration, represented as \( C \), is crucial in indefinite integrals. Whenever you find an antiderivative, the constant \( C \) signifies that the solution is not unique.
The reason for \( C \) stems from the fact that differentiating any constant gives zero. Hence, when finding antiderivatives, the opposite is true; many original functions could have led to the same derivative.
The reason for \( C \) stems from the fact that differentiating any constant gives zero. Hence, when finding antiderivatives, the opposite is true; many original functions could have led to the same derivative.
- This concept assures that every possible function, differing by a constant, is accounted for in the antiderivative family.
- In practical problems, an initial condition or additional data help specify the exact value of \( C \), making the solution particular to a scenario.
Other exercises in this chapter
Problem 29
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