### Step-by-Step Solution
#### 1. Simplify the Integrand
We begin by simplifying the integrand [tex]\( \frac{\sqrt[4]{x^5}}{x^{7/2}} \)[/tex].
First, express [tex]\( \sqrt[4]{x^5} \)[/tex] in terms of exponents:
[tex]\[
\sqrt[4]{x^5} = x^{5/4}
\][/tex]
Now, rewrite the integrand:
[tex]\[
\frac{x^{5/4}}{x^{7/2}}
\][/tex]
Using the rule of exponents [tex]\( \frac{a^m}{a^n} = a^{m-n} \)[/tex], we get:
[tex]\[
\frac{x^{5/4}}{x^{7/2}} = x^{5/4 - 7/2}
\][/tex]
Convert the exponents to have a common denominator:
[tex]\[
5/4 = \frac{5}{4} \quad \text{and} \quad 7/2 = \frac{7 \times 2}{2 \times 2} = \frac{14}{4}
\][/tex]
So:
[tex]\[
x^{5/4 - 14/4} = x^{\frac{5 - 14}{4}} = x^{-9/4}
\][/tex]
Thus, our integrand is:
[tex]\[
\frac{1}{x^{9/4}}
\][/tex]
#### 2. Perform the Integration
Now, we need to find the indefinite integral of the simplified integrand:
[tex]\[
\int x^{-9/4} \, dx
\][/tex]
We use the power rule for integration:
[tex]\[
\int x^n \, dx = \frac{x^{n+1}}{n+1} + C
\][/tex]
Here, [tex]\( n = -9/4 \)[/tex]:
[tex]\[
n+1 = -9/4 + 4/4 = -5/4
\][/tex]
Thus:
[tex]\[
\int x^{-9/4} \, dx = \frac{x^{-5/4}}{-5/4} + C = -\frac{4}{5} x^{-5/4} + C
\][/tex]
We simplify this to:
[tex]\[
-\frac{4}{5} x^{-5/4} + C
\][/tex]
#### 3. Check by Differentiation
To verify our result, we differentiate [tex]\( -\frac{4}{5} x^{-5/4} + C \)[/tex].
The derivative of [tex]\( C \)[/tex] is 0.
Differentiate [tex]\( -\frac{4}{5} x^{-5/4} \)[/tex]:
Using the power rule for differentiation [tex]\( \frac{d}{dx}[x^n] = n x^{n-1} \)[/tex]:
Here, [tex]\( n = -5/4 \)[/tex]:
[tex]\[
\frac{d}{dx}\left( -\frac{4}{5} x^{-5/4} \right) = -\frac{4}{5} \cdot (-5/4) x^{-5/4 - 1} = x^{-9/4}
\][/tex]
The constants [tex]\( -\frac{4}{5} \cdot (-5/4) \)[/tex] cancel out as follows:
[tex]\[
-\frac{4}{5} \cdot (-5/4) = 1
\][/tex]
So we get:
[tex]\[
x^{-9/4}
\][/tex]
This confirms that our integration result is correct.
### Final Answer
The indefinite integral of the given function is:
[tex]\[
-\frac{4}{5} x^{-5/4} + C
\][/tex]
When differentiated, this yields the original integrand:
[tex]\[
x^{-9/4} + C
\][/tex]
