Question

integrate using u substitution:
(e^√x)/(√x)

Answer 1

\[\frac{ e ^{\sqrt{x}} }{ \sqrt{x} }\]

Answer 2

so first we have to decide our u

Answer 3

This will come to practice if you are just now starting u substitution

Answer 4

From what I see the best thing to do would be to make \[u=\sqrt{x}\]

Answer 5

using the power rule as x^(1/2) we get\[du=\frac{ 1 }{ 2\sqrt{x} }\]

Answer 6

how do we get from the derivative to the e function?

Answer 7

I dont fully understand what you're asking? what exactly is troubling you?

Answer 8

If I think I get what you're saying you don't fully understand how we go from integrating to using a derivative

Answer 9

well lets write out the full integral\[\int\limits_{}^{}\frac{ e ^{\sqrt{x}} }{ \sqrt{x} }dx\]

Answer 10

well, when we find dy/dx, arent we supposed to find a way to make dy/dx equal the other term?
Say the function was xsin(2x).
u=x^2 dy/dx=2x therefore x=1/2(dy/dx)

Answer 11

*xsin(x^2) is the original function sorry

Answer 12

now with our understanding of the rules of integration we cannot solve this. This is why we use substitution. We find a way to replace the variable that is solveable with our knowledge

Answer 13

so first we must replace the x with something in this case we choose u but we relate u to x by making it equal to x in some way hence
u=sqrt(x)

Answer 14

next we take the derivative with respect to x so we have\[\frac{ du }{ dx }=\frac{ 1 }{ 2\sqrt{x} }\]

Answer 15

so we have to find a way to make 1/(2√x)=e^√x right?

Answer 16

oops

Answer 17

we have to make the derivative equal to the other function, no?

Answer 18

now what we our finding in our original equation is replacing dx with du because in the original integral we have to variables x and u but we want to get it in terms of 1 variable\[\int\limits_{}^{}\frac{ e ^{u}} { \sqrt{x} }dx\]

Answer 19

so what we are solving for is that dx and is why when we chose u we then take the deriivative of it with respect to x because that gives us du/dx

Answer 20

If you let \(u = \sqrt{x}\), as @recon14193 mentioned, then You could resubstitute that into your original function:\[\frac{e^{u}}{u}\]Now just integrate \[u=\sqrt{x} ~ , ~ du = \frac{1}{2\sqrt{x}} \iff 2du =\frac{1}{\sqrt{x}}\]\[=2\int e^udu\] I believe....

Answer 21

multiplying both sides by dx gives a relation of du to dx so\[du=\frac{ 1 }{ 2\sqrt{x} }dx\]

Answer 22

that's correct @Jhannybean

Answer 23

where did the sqrtx from the bottom go?

Answer 24

you replaced them with u because we made \[u=\sqrt{x}\]

Answer 25

so where we see the sqrtx we plug in a u

Answer 26

the final answer is 2e^sqrtx +c...i still dont understand where the u from the denominator went