# 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} 