Looking for trustworthy answers? Westonci.ca is the ultimate Q&A platform where experts share their knowledge on various topics. Discover comprehensive answers to your questions from knowledgeable professionals on our user-friendly platform. Experience the convenience of finding accurate answers to your questions from knowledgeable experts on our platform.
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]
Thank you for trusting us with your questions. We're here to help you find accurate answers quickly and efficiently. Thank you for your visit. We're committed to providing you with the best information available. Return anytime for more. Find reliable answers at Westonci.ca. Visit us again for the latest updates and expert advice.