How do you integrate #e^sqrt(z) /sqrt(z) dz#?

Answer 1

Ezra Adams

#2 e^sqrt z +C#

Use #(sqrt z)'=1/2(1/sqrtz)#

Here,

#int e^sqrt z/sqrt z dz#

#=2 int e^sqrt z (sqrt z)' dz#

#=2 int e^sqrt z d(sqrt z)#

#=2 e^sqrt z+C#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Emily Ammerman

To integrate ( \frac{e^{\sqrt{z}}}{\sqrt{z}} , dz ), we can employ a substitution method. Let ( u = \sqrt{z} ), then ( z = u^2 ) and ( dz = 2u , du ). Substituting these into the integral, we get:

[ \int \frac{e^u}{u} \cdot 2u , du ]

Simplifying, we have:

[ 2 \int e^u , du ]

Integrating ( e^u ) with respect to ( u ) gives:

[ 2e^u + C ]

Substituting back ( u = \sqrt{z} ), we obtain the final result:

[ 2e^{\sqrt{z}} + C ]

So, the integral of ( \frac{e^{\sqrt{z}}}{\sqrt{z}} , dz ) is ( 2e^{\sqrt{z}} + C ), where ( C ) is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.