use the substitution of u = √x, ∫₁⁴ (e^√x / √x) dx is equal to which of the following?\no a 2 ∫₁² e^u du\no…

use the substitution of u = √x, ∫₁⁴ (e^√x / √x) dx is equal to which of the following?\no a 2 ∫₁² e^u du\no b ∫₁⁴ e^u du\no c 1/2 ∫₁² e^u du\no d 2 ∫₁¹⁶ e^u du\no e 2 ∫₁⁴ e^u du

use the substitution of u = √x, ∫₁⁴ (e^√x / √x) dx is equal to which of the following?\no a 2 ∫₁² e^u du\no b ∫₁⁴ e^u du\no c 1/2 ∫₁² e^u du\no d 2 ∫₁¹⁶ e^u du\no e 2 ∫₁⁴ e^u du

Answer

Explanation:

Step1: Find $dx$ in terms of $du$

Given $u = \sqrt{x}$, then $x=u^{2}$ and $dx = 2u\ du$.

Step2: Change the limits of integration

When $x = 1$, $u=\sqrt{1}=1$; when $x = 4$, $u=\sqrt{4}=2$.

Step3: Substitute into the integral

The integral $\int_{1}^{4}\frac{e^{\sqrt{x}}}{\sqrt{x}}dx$ becomes $\int_{1}^{2}\frac{e^{u}}{u}\cdot2u\ du=2\int_{1}^{2}e^{u}du$.

Answer:

A. $2\int_{1}^{2}e^{u}du$