Question

Evaluate the integral\(\int\limits_1^4 {{e^{\sqrt x }}dx} \)

Short answer

We will use the substitute method and integral by parts method to solve the integral

Let \(w = \sqrt x \)

Step-by-step solution

Given Data

Given =\({\rm I} = \int\limits_1^4 {{e^{\sqrt x }}dx} \)

\(\therefore {\rm I} = \int\limits_1^2 {{e^w}2wdw} \)

 Step 2: Using Integration by Part Method

\({\rm I} = \mu v - \int {vd\mu } \)

Let,\(\begin{aligned}{l}\mu = 2w\\d\mu = 2dw\end{aligned}\) \(\begin{aligned}{l}dv = {e^w}dw\\v = {e^w}\end{aligned}\)

Substituting the above equation

\(\begin{aligned}{l}{\rm I} = \left( {2w{e^w}} \right)_1^2 - 2\int\limits_1^2 {{e^w}dw} \\{\rm I} = 4{e^2} - 2e - 2\left( {{e^w}} \right)_1^2\\{\rm I} = 4{e^2} - 2e - 2\left( {{e^2} - e} \right)\\{\rm I} = 4{e^2} - 2e - 2{e^2} - 2e\\{\rm I} = 2{e^2}\end{aligned}\)

Hence, \(\int\limits_1^4 {{e^{\sqrt x }}dx} = 2{e^2}\)