Find the best solutions to your problems with the help of IDNLearn.com. Ask your questions and receive reliable and comprehensive answers from our dedicated community of professionals.

Find the indefinite integral and check the result by differentiation. (Use [tex]$C$[/tex] for the constant of integration.)

[tex]\[ \int \left(x^{3/2} + 8x + 8 \right) \, dx \][/tex]


Sagot :

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.