Evaluate the integral Evaluate the integral.

{eq}\int_0^5 e^{-x} + 4 dx {/eq}

Question:

Evaluate the integral Evaluate the integral.

{eq}\int_0^5 e^{-x} + 4 dx {/eq}

Definite Integral in Calculus:

The definite integral is used to compute an exact area. If a function {eq}f(x) {/eq} is continuous on the interval {eq}\left[ a,b \right] {/eq} then the definite integral of {eq}f(x) {/eq} from {eq}a {/eq} to {eq}b {/eq} is {eq}\displaystyle \int_{a}^{b} f(x) \ dx {/eq}.

To solve this problem, we'll use the integral sum rule {eq}\displaystyle \int \left( f(x) + g(x) \right) \, \mathrm{d}x = \int f(x) \ dx + \int g(x) \, \mathrm{d}x {/eq}.

Next, we'll use the common integral {eq}\displaystyle \int e^{ax} \mathrm{d}x = \dfrac{e^{ax}}{a}+C {/eq} and plug in the boundaries to get the desired solution.

Answer and Explanation: 1

We are given:

{eq}\displaystyle \int_0^5 (e^{-x} + 4) dx {/eq}

Compute the indefinite integral:

{eq}= \displaystyle \int (e^{-x} + 4 )dx {/eq}

Apply integral sum rule:

{eq}= \displaystyle \int e^{-x} \ dx + \int 4 \ dx {/eq}

{eq}= \displaystyle \dfrac{e^{-x} }{-1} + 4x+C \quad \text{Where C is an arbitrary constant.)} {/eq}

{eq}= \displaystyle -e^{-x} + 4x+C {/eq}

Plug in the boundaries:

{eq}= \displaystyle -e^{-5} + 20+e^0 -0 {/eq}

{eq}= \displaystyle -e^{-5} + 21 {/eq}

Therefore the solution is:

{eq}\displaystyle \int_0^5 (e^{-x} + 4) dx = \displaystyle -e^{-5} + 21. {/eq}


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Evaluating Definite Integrals Using the Fundamental Theorem

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Chapter 16 / Lesson 2
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In calculus, the fundamental theorem is an essential tool that helps explain the relationship between integration and differentiation. Learn about evaluating definite integrals using the fundamental theorem, and work examples to gain understanding.


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