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Math  /  Calculus

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6. [-/1 Points]
DETAILS MY NOTES SCALCET9 4.9.021.
Find the most general antiderivative of the function. (Check your answer by differentiation. Use CC for the constant of the antiderivative.) f(θ)=4sin⁡(θ)−9sec⁡(θ)tan⁡(θ)f(\theta)=4 \sin (\theta)-9 \sec (\theta) \tan (\theta) \quad on the interval (−π2,π2)\left(-\frac{\pi}{2}, \frac{\pi}{2}\right) F(θ)=F(\theta)= □\square
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Studdy Solution

STEP 1

1. We are given the function f(θ)=4sin⁡(θ)−9sec⁡(θ)tan⁡(θ) f(\theta) = 4 \sin(\theta) - 9 \sec(\theta) \tan(\theta) .
2. We need to find the most general antiderivative F(θ) F(\theta) .
3. The antiderivative will include a constant of integration denoted by C C .

STEP 2

1. Identify the antiderivative of each term in the function separately.
2. Combine the antiderivatives to form the general antiderivative.
3. Verify the antiderivative by differentiating it.
4. Include the constant of integration C C .

STEP 3

Identify the antiderivative of 4sin⁡(θ) 4 \sin(\theta) :
The antiderivative of sin⁡(θ) \sin(\theta) is −cos⁡(θ)-\cos(\theta).
Thus, the antiderivative of 4sin⁡(θ) 4 \sin(\theta) is:
−4cos⁡(θ) -4 \cos(\theta)

STEP 4

Identify the antiderivative of −9sec⁡(θ)tan⁡(θ) -9 \sec(\theta) \tan(\theta) :
The antiderivative of sec⁡(θ)tan⁡(θ) \sec(\theta) \tan(\theta) is sec⁡(θ) \sec(\theta) .
Thus, the antiderivative of −9sec⁡(θ)tan⁡(θ) -9 \sec(\theta) \tan(\theta) is:
−9sec⁡(θ) -9 \sec(\theta)

STEP 5

Combine the antiderivatives from STEP_1 and STEP_2:
F(θ)=−4cos⁡(θ)−9sec⁡(θ)+C F(\theta) = -4 \cos(\theta) - 9 \sec(\theta) + C

STEP 6

Verify by differentiating F(θ) F(\theta) :
Differentiate F(θ)=−4cos⁡(θ)−9sec⁡(θ)+C F(\theta) = -4 \cos(\theta) - 9 \sec(\theta) + C :
ddθ(−4cos⁡(θ))=4sin⁡(θ) \frac{d}{d\theta}(-4 \cos(\theta)) = 4 \sin(\theta) ddθ(−9sec⁡(θ))=−9sec⁡(θ)tan⁡(θ) \frac{d}{d\theta}(-9 \sec(\theta)) = -9 \sec(\theta) \tan(\theta) ddθ(C)=0 \frac{d}{d\theta}(C) = 0
The derivative is:
4sin⁡(θ)−9sec⁡(θ)tan⁡(θ) 4 \sin(\theta) - 9 \sec(\theta) \tan(\theta)
This matches the original function f(θ) f(\theta) .

STEP 7

Include the constant of integration C C in the final answer:
The most general antiderivative is:
F(θ)=−4cos⁡(θ)−9sec⁡(θ)+C F(\theta) = -4 \cos(\theta) - 9 \sec(\theta) + C
The most general antiderivative is:
−4cos⁡(θ)−9sec⁡(θ)+C \boxed{-4 \cos(\theta) - 9 \sec(\theta) + C}

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