Answer to Question #162053 in Calculus for Phyroe

Question

Evaluate the integral of (e^(√x)/√x) dx from 1 to 4.

Answer 1

Evaluate $\int$ (e^($\sqrt{x}$ )/$\sqrt{x}$ ) dx from 1 to 4

Solution

let v = $\sqrt{x}$ ; hence dv = d/dx ($\sqrt{x}$ ) = 1/(2$\sqrt{x}$ ) dx

So $\int$ (e^($\sqrt{x}$ )/$\sqrt{x}$ ) dx = $\int$ 2e^v dv

Applying constant multiple rule: $\int$ c f(v) dv = c$\int$ f(v) dv with c = 2 and f(v) = e^v

$\int$ 2e^v dv = 2$\int$ e^v dv

The integral of the exponential function is: $\int$ e^v dv = e^v

Therefore: 2$\int$ e^v dv = 2e^v

Recall that v = $\sqrt{x}$

Therefore $\int$ (e^($\sqrt{x}$ )/$\sqrt{x}$ ) dx = 2e^($\sqrt{x}$ )

Applying the limits from 1 to 4 to the integral result we get:

2e^($\sqrt{4}$ ) - 2e^$(\sqrt{1})$ = 9.341548541

Answer: 9.341548541