To solve the given problem, we need to find the indefinite integral of the function [tex]\( x^{3/2} + 8x + 8 \)[/tex], and then verify the result by differentiating the integral.
### Step 1: Set up the integral
Consider the integral:
[tex]\[
\int \left(x^{3/2} + 8x + 8\right) \, dx
\][/tex]
### Step 2: Integrate each term separately
We will integrate each term of the integrand separately.
1. Integrate [tex]\( x^{3/2} \)[/tex]:
[tex]\[
\int x^{3/2} \, dx
\][/tex]
Use the power rule for integration: [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex].
Here, [tex]\( n = \frac{3}{2} \)[/tex]:
[tex]\[
\int x^{3/2} \, dx = \frac{x^{3/2 + 1}}{3/2 + 1} = \frac{x^{5/2}}{5/2} = \frac{2}{5} x^{5/2}
\][/tex]
2. Integrate [tex]\( 8x \)[/tex]:
[tex]\[
\int 8x \, dx
\][/tex]
Use the power rule for integration: [tex]\(\int x^n \, dx = \frac{x^{n+1}}{n+1} + C\)[/tex].
Here, [tex]\( n = 1 \)[/tex]:
[tex]\[
\int 8x \, dx = 8 \left( \frac{x^{1+1}}{1+1} \right) = 8 \left( \frac{x^2}{2} \right) = 4x^2
\][/tex]
3. Integrate [tex]\( 8 \)[/tex]:
[tex]\[
\int 8 \, dx
\][/tex]
The integral of a constant is the constant multiplied by the variable:
[tex]\[
\int 8 \, dx = 8x
\][/tex]
### Step 3: Combine the results of the integrals
Now sum up the integrated results from each term:
[tex]\[
\int \left( x^{3/2} + 8x + 8 \right) \, dx = \frac{2}{5} x^{5/2} + 4x^2 + 8x + C
\][/tex]
Thus, the indefinite integral of the function is:
[tex]\[
\boxed{\frac{2}{5} x^{5/2} + 4x^2 + 8x + C}
\][/tex]
### Step 4: Verify by differentiation
We now differentiate the result to check if we recover the original integrand. We need to differentiate:
[tex]\[
\frac{2}{5} x^{5/2} + 4x^2 + 8x + C
\][/tex]
1. Differentiate [tex]\( \frac{2}{5} x^{5/2} \)[/tex]:
Use the power rule for differentiation: [tex]\(\frac{d}{dx}(x^n) = nx^{n-1}\)[/tex].
[tex]\[
\frac{d}{dx} \left( \frac{2}{5} x^{5/2} \right) = \frac{2}{5} \cdot \frac{5}{2} x^{5/2-1} = x^{3/2}
\][/tex]
2. Differentiate [tex]\( 4x^2 \)[/tex]:
[tex]\[
\frac{d}{dx} \left( 4x^2 \right) = 4 \cdot 2x = 8x
\][/tex]
3. Differentiate [tex]\( 8x \)[/tex]:
[tex]\[
\frac{d}{dx} \left( 8x \right) = 8
\][/tex]
4. Differentiate [tex]\( C \)[/tex] (a constant):
[tex]\[
\frac{d}{dx} (C) = 0
\][/tex]
### Step 5: Combine the differentiated results
Combining all the results from the differentiation, we get:
[tex]\[
x^{3/2} + 8x + 8
\][/tex]
This is indeed our original integrand, verifying that our integration was correct.
Thus, the integral [tex]\( \frac{2}{5} x^{5/2} + 4x^2 + 8x + C \)[/tex] is correctly confirmed by differentiation.
