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Definite Integrals of Exponential FunctionsDefinite Integrals of Exponential Functions
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Question 1 of 2
1. Question
Find the integral$$\int_{-1}^{3} e^{-\frac{x}{2}} dx$$Hint
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Integrating Exponential Functions with Base “e”
$$\int e^{\color{#004ec4}{a}x+b} dx=\frac{1}{\color{#004ec4}{a}} e^{\color{#004ec4}{a}x+b} +c$$Substitute the components into the formula$$\int e^{\color{#004ec4}{a}x+b} dx$$ `=` $$\frac{1}{\color{#004ec4}{a}} e^{\color{#004ec4}{a}x+b} +c$$ $$\int_{-1}^{3} e^{\color{#004ec4}{-\frac{x}{2}}} dx$$ `=` $$\int_{-1}^{3} \frac{e^{\color{#004ec4}{-\frac{x}{2}}}}{-\frac{1}{2}}$$ Substitute known values `=` `[-2e^(-x/2)]_(-1)^3` Simplify Finally, get the difference of the upper and lower limits substituted to the integral as `x`.`[-2e^(-x/2)]_(-1)^3` `=` `[-2e^(-3/2)]-[-2e^(- (-1)/2)]` Substitute the limits `=` `-2e^(-3/2)+2e^(1/2)` Evaluate `=` `-2/e^(3/2) +2e^(1/2)` Reciprocate `e^(-3/2)` `=` `-2/(sqrt(e^3)) +2sqrte` Change the exponents into surds `-2/(sqrt(e^3)) +2sqrte` -
Question 2 of 2
2. Question
Find the integral$$\int_{1}^{2} 5e^{2x-1} dx$$Hint
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Integrating Exponential Functions with Base “e”
$$\int e^{\color{#004ec4}{a}x+b} dx=\frac{1}{\color{#004ec4}{a}} e^{\color{#004ec4}{a}x+b} +c$$Substitute the components into the formula$$\int e^{\color{#004ec4}{a}x+b} dx$$ `=` $$\frac{1}{\color{#004ec4}{a}} e^{\color{#004ec4}{a}x+b} +c$$ $$\int_{1}^{2} 5e^{\color{#004ec4}{2}x-1} dx$$ `=` $$\left[\frac{5e^{\color{#004ec4}{2}x-1}}{2}\right]_{1}^{2}$$ Substitute known values Finally, get the difference of the upper and lower limits substituted to the integral as `x`.`[(5e^(2x-1))/2]_1^2` `=` `[(5e^(2(2)-1))/2]-[(5e^(2(1)-1))/2]` Substitute the limits `=` `(5e^3)/2-(5e)/2` Evaluate `=` `5/2 e[e^2-1]` Factorise `5/2 e[e^2-1]`
Quizzes
- Antiderivatives (Indefinite Integrals) 1
- Antiderivatives (Indefinite Integrals) 2
- Antiderivatives (Indefinite Integrals) 3
- Antiderivatives of Exponential Functions
- Antiderivatives of Logarithmic Functions 1
- Antiderivatives of Logarithmic Functions 2
- Antiderivatives of Trig Functions 1
- Antiderivatives of Trig Functions 2
- Definite Integrals
- Definite Integrals of Exponential Functions
- Definite Integrals of Logarithmic Functions