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Sagot :
Let's solve the given indefinite integral step-by-step and verify our result by differentiating the antiderivative.
We need to find:
[tex]\[ \int\left(9 x^{17} - 3 x^5\right) dx \][/tex]
### Step 1: Split the Integral
First, split the integral into two separate integrals:
[tex]\[ \int\left(9 x^{17} - 3 x^5\right) dx = \int 9 x^{17} \, dx - \int 3 x^5 \, dx \][/tex]
### Step 2: Integrate Each Term
Now, integrate each term separately.
#### Integrate [tex]\(9 x^{17}\)[/tex]:
[tex]\[ \int 9 x^{17} \, dx = 9 \int x^{17} \, dx \][/tex]
We use the power rule for integration, which states:
[tex]\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \][/tex]
In this case, [tex]\( n = 17 \)[/tex], so:
[tex]\[ 9 \int x^{17} \, dx = 9 \left( \frac{x^{18}}{18} \right) = \frac{9}{18} x^{18} = \frac{1}{2} x^{18} \][/tex]
#### Integrate [tex]\(3 x^5\)[/tex]:
[tex]\[ \int 3 x^5 \, dx = 3 \int x^5 \, dx \][/tex]
Again, using the power rule with [tex]\( n = 5 \)[/tex]:
[tex]\[ 3 \int x^5 \, dx = 3 \left( \frac{x^6}{6} \right) = \frac{3}{6} x^6 = \frac{1}{2} x^6 \][/tex]
### Step 3: Combine the Results
Now, combine the results from the two integrals:
[tex]\[ \int\left(9 x^{17} - 3 x^5\right) dx = \frac{1}{2} x^{18} - \frac{1}{2} x^6 + C \][/tex]
Here, [tex]\( C \)[/tex] is the constant of integration.
So the indefinite integral is:
[tex]\[ \int\left(9 x^{17} - 3 x^5\right) dx = \frac{1}{2} x^{18} - \frac{1}{2} x^6 + C \][/tex]
### Step 4: Verify by Differentiation
To verify, we differentiate the result:
[tex]\[ \frac{d}{dx} \left( \frac{1}{2} x^{18} - \frac{1}{2} x^6 + C \right) \][/tex]
Using the power rule for differentiation [tex]\(\frac{d}{dx} x^n = n x^{n-1}\)[/tex]:
[tex]\[ \frac{d}{dx} \left( \frac{1}{2} x^{18} \right) = \frac{1}{2} \cdot 18 x^{17} = 9 x^{17} \][/tex]
[tex]\[ \frac{d}{dx} \left( \frac{1}{2} x^6 \right) = \frac{1}{2} \cdot 6 x^5 = 3 x^5 \][/tex]
[tex]\[ \frac{d}{dx} (C) = 0 \][/tex]
Combining these, we get:
[tex]\[ \frac{d}{dx} \left( \frac{1}{2} x^{18} - \frac{1}{2} x^6 + C \right) = 9 x^{17} - 3 x^5 \][/tex]
This matches the original integrand, so our solution is verified.
Thus, the indefinite integral is:
[tex]\[ \int\left(9 x^{17} - 3 x^5\right) dx = \frac{1}{2} x^{18} - \frac{1}{2} x^6 + C \][/tex]
We need to find:
[tex]\[ \int\left(9 x^{17} - 3 x^5\right) dx \][/tex]
### Step 1: Split the Integral
First, split the integral into two separate integrals:
[tex]\[ \int\left(9 x^{17} - 3 x^5\right) dx = \int 9 x^{17} \, dx - \int 3 x^5 \, dx \][/tex]
### Step 2: Integrate Each Term
Now, integrate each term separately.
#### Integrate [tex]\(9 x^{17}\)[/tex]:
[tex]\[ \int 9 x^{17} \, dx = 9 \int x^{17} \, dx \][/tex]
We use the power rule for integration, which states:
[tex]\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \][/tex]
In this case, [tex]\( n = 17 \)[/tex], so:
[tex]\[ 9 \int x^{17} \, dx = 9 \left( \frac{x^{18}}{18} \right) = \frac{9}{18} x^{18} = \frac{1}{2} x^{18} \][/tex]
#### Integrate [tex]\(3 x^5\)[/tex]:
[tex]\[ \int 3 x^5 \, dx = 3 \int x^5 \, dx \][/tex]
Again, using the power rule with [tex]\( n = 5 \)[/tex]:
[tex]\[ 3 \int x^5 \, dx = 3 \left( \frac{x^6}{6} \right) = \frac{3}{6} x^6 = \frac{1}{2} x^6 \][/tex]
### Step 3: Combine the Results
Now, combine the results from the two integrals:
[tex]\[ \int\left(9 x^{17} - 3 x^5\right) dx = \frac{1}{2} x^{18} - \frac{1}{2} x^6 + C \][/tex]
Here, [tex]\( C \)[/tex] is the constant of integration.
So the indefinite integral is:
[tex]\[ \int\left(9 x^{17} - 3 x^5\right) dx = \frac{1}{2} x^{18} - \frac{1}{2} x^6 + C \][/tex]
### Step 4: Verify by Differentiation
To verify, we differentiate the result:
[tex]\[ \frac{d}{dx} \left( \frac{1}{2} x^{18} - \frac{1}{2} x^6 + C \right) \][/tex]
Using the power rule for differentiation [tex]\(\frac{d}{dx} x^n = n x^{n-1}\)[/tex]:
[tex]\[ \frac{d}{dx} \left( \frac{1}{2} x^{18} \right) = \frac{1}{2} \cdot 18 x^{17} = 9 x^{17} \][/tex]
[tex]\[ \frac{d}{dx} \left( \frac{1}{2} x^6 \right) = \frac{1}{2} \cdot 6 x^5 = 3 x^5 \][/tex]
[tex]\[ \frac{d}{dx} (C) = 0 \][/tex]
Combining these, we get:
[tex]\[ \frac{d}{dx} \left( \frac{1}{2} x^{18} - \frac{1}{2} x^6 + C \right) = 9 x^{17} - 3 x^5 \][/tex]
This matches the original integrand, so our solution is verified.
Thus, the indefinite integral is:
[tex]\[ \int\left(9 x^{17} - 3 x^5\right) dx = \frac{1}{2} x^{18} - \frac{1}{2} x^6 + C \][/tex]
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