Question

evaluate the integral.
\int \frac{e^{8\sqrt{z}}}{\sqrt{z}} \\, dz

\int \frac{e^{8\sqrt{z}}}{\sqrt{z}} \\, dz = \square

Explanation:

Step1: Use substitution method

Let \( u = 8\sqrt{z}=8z^{\frac{1}{2}} \), then find the derivative of \( u \) with respect to \( z \).
The derivative \( du=\frac{d}{dz}(8z^{\frac{1}{2}})dz = 8\times\frac{1}{2}z^{-\frac{1}{2}}dz=\frac{4}{\sqrt{z}}dz \), so \( \frac{1}{\sqrt{z}}dz=\frac{du}{4} \).

Step2: Rewrite the integral

Substitute \( u \) and \( \frac{1}{\sqrt{z}}dz \) into the original integral \( \int\frac{e^{8\sqrt{z}}}{\sqrt{z}}dz \), we get:
\( \int e^{u}\times\frac{du}{4}=\frac{1}{4}\int e^{u}du \)

Step3: Integrate \( e^{u} \)

We know that the integral of \( e^{u} \) with respect to \( u \) is \( e^{u}+C \) (where \( C \) is the constant of integration). So \( \frac{1}{4}\int e^{u}du=\frac{1}{4}e^{u}+C \)

Step4: Substitute back \( u \)

Substitute \( u = 8\sqrt{z} \) back into the expression, we get \( \frac{1}{4}e^{8\sqrt{z}}+C \) (If we consider a definite integral, but since the problem doesn't specify limits, we assume it's an indefinite integral. However, if we assume it's a definite integral with limits (maybe from some value, but since not given, maybe a typo and it's an indefinite integral or maybe from 0 to some value, but as per the problem as given, we proceed with indefinite integral first. Wait, maybe the original problem has limits? Wait the user's image shows the integral without limits? Wait no, maybe it's a definite integral but the limits are missing? Wait no, looking at the image again, maybe it's an indefinite integral. Wait, but let's check again. Wait, maybe the integral is from, say, 0 to some value, but the problem as presented has no limits. Wait, maybe it's a typo and the integral is \( \int\frac{e^{8\sqrt{z}}}{\sqrt{z}}dz \) (indefinite) or maybe with limits. Wait, but the user's problem shows the integral with a square box, maybe expecting an antiderivative. Wait, let's re - check the substitution.

Wait, when we do substitution for \( \int\frac{e^{8\sqrt{z}}}{\sqrt{z}}dz \), let \( u = 8\sqrt{z} \), then \( du=\frac{4}{\sqrt{z}}dz\implies\frac{1}{\sqrt{z}}dz=\frac{du}{4} \). Then the integral becomes \( \int e^{u}\times\frac{du}{4}=\frac{1}{4}e^{u}+C=\frac{1}{4}e^{8\sqrt{z}}+C \).

But maybe the problem is a definite integral with lower limit 0? Wait, if we assume the lower limit is 0 (since when \( z = 0 \), \( u=0 \)) and upper limit such that when we integrate, but the problem as given has no limits. Wait, maybe the original problem has a typo and the integral is \( \int_{0}^{z}\frac{e^{8\sqrt{t}}}{\sqrt{t}}dt \) or maybe it's an indefinite integral. Wait, the user's problem shows \( \int\frac{e^{8\sqrt{z}}}{\sqrt{z}}dz=\square \), so we can give the antiderivative.

Wait, but let's check the calculation again. The derivative of \( e^{8\sqrt{z}} \) with respect to \( z \) is \( e^{8\sqrt{z}}\times\frac{d}{dz}(8\sqrt{z})=e^{8\sqrt{z}}\times\frac{4}{\sqrt{z}} \). So if we have \( \int\frac{e^{8\sqrt{z}}}{\sqrt{z}}dz \), we can see that \( \frac{d}{dz}(\frac{1}{4}e^{8\sqrt{z}})=\frac{1}{4}\times e^{8\sqrt{z}}\times\frac{4}{\sqrt{z}}=\frac{e^{8\sqrt{z}}}{\sqrt{z}} \), which matches the integrand. So the antiderivative is \( \frac{1}{4}e^{8\sqrt{z}}+C \). But if we assume it's a definite integral from 0 to \( z \) (or maybe the problem is missing limits, but if we consider the integral as an indefinite integral, the answer is \( \frac{1}{4}e^{8\sqrt{z}}+C \). But maybe the problem was supposed to have limits, like from 0 to 1? Wait, no, the user's problem as given has no limits. Wait, maybe it's a mistake and the integral is \( \int\frac{e^{8\sqrt{z}}}{\sqrt{z}}dz \) (indefinite), so the answer is \( \frac{1}{4}e^{8\sqrt{z}}+C \). But if we assume that it's a definite integral with lower limit 0 (since when \( z = 0 \), the integrand is defined in a limit sense) and upper limit, say, \( z \), but no, the problem is written as \( \int\frac{e^{8\sqrt{z}}}{\sqrt{z}}dz=\square \). So we can present the antiderivative.

Answer:

\( \frac{1}{4}e^{8\sqrt{z}}+C \) (If it's an indefinite integral. If it's a definite integral with, for example, lower limit 0 and upper limit \( z \), but as per the problem statement, the most appropriate answer for the integral \( \int\frac{e^{8\sqrt{z}}}{\sqrt{z}}dz \) is \( \frac{1}{4}e^{8\sqrt{z}}+C \))