Question
Find the most general antiderivative of the function. \[ f(\theta)=\sec (\theta) \tan (\theta)-5 e^{\theta} \] \( f(\theta)=\square \)
Ask by Little Nguyen. in the United States
Mar 19,2025
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Answer
The most general antiderivative of \( f(\theta) = \sec(\theta) \tan(\theta) - 5e^{\theta} \) is \( \sec(\theta) - 5e^{\theta} + C \), where \( C \) is a constant.
Solution
Find the antiderivative of \( \sec(\theta)\tan(\theta)-5e^{\theta} \).
Evaluate the integral by following steps:
- step0: Evaluate using formulas and rules:
\(\int \left(\sec\left(\theta \right)\tan\left(\theta \right)-5e^{\theta }\right) d\theta\)
- step1: Use properties of integrals:
\(\int \sec\left(\theta \right)\tan\left(\theta \right) d\theta+\int -5e^{\theta } d\theta\)
- step2: Evaluate the integral:
\(\sec\left(\theta \right)+\int -5e^{\theta } d\theta\)
- step3: Evaluate the integral:
\(\sec\left(\theta \right)-5e^{\theta }\)
- step4: Add the constant of integral C:
\(\sec\left(\theta \right)-5e^{\theta } + C, C \in \mathbb{R}\)
The most general antiderivative of the function \( f(\theta) = \sec(\theta) \tan(\theta) - 5e^{\theta} \) is \( \sec(\theta) - 5e^{\theta} + C \), where \( C \) is an arbitrary constant.
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Simplify this solution Extra Insights
To find the most general antiderivative of the function \( f(\theta) = \sec(\theta) \tan(\theta) - 5 e^{\theta} \), we can integrate each term separately. The integral of \( \sec(\theta) \tan(\theta) \) is \( \sec(\theta) + C_1 \), where \( C_1 \) is a constant of integration. The integral of \( -5 e^{\theta} \) is \( -5 e^{\theta} + C_2 \), where \( C_2 \) is another constant. Combining these results, we have: \[ \int f(\theta) \, d\theta = \sec(\theta) - 5 e^{\theta} + C \] where \( C = C_1 + C_2 \) is the most general constant. Thus, the most general antiderivative is: \[ F(\theta) = \sec(\theta) - 5 e^{\theta} + C \]
