At Westonci.ca, we connect you with experts who provide detailed answers to your most pressing questions. Start exploring now! Our Q&A platform provides quick and trustworthy answers to your questions from experienced professionals in different areas of expertise. Discover detailed answers to your questions from a wide network of experts on our comprehensive Q&A platform.
Sagot :
Sure, let's go through the steps together to find the antiderivative [tex]\( F(x) \)[/tex] of the function [tex]\( f(x) = \frac{6}{x^3} - \frac{6}{x^7} \)[/tex], given the initial condition [tex]\( F(1) = 0 \)[/tex].
1. Express the function in a more integrable form:
First, rewrite [tex]\( f(x) \)[/tex] with negative exponents:
[tex]\[ f(x) = 6x^{-3} - 6x^{-7} \][/tex]
2. Find the antiderivative:
To find the antiderivative [tex]\( F(x) \)[/tex], we integrate [tex]\( f(x) \)[/tex] term by term:
[tex]\[ F(x) = \int (6x^{-3} - 6x^{-7}) \, dx \][/tex]
Using the power rule for integration [tex]\( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)[/tex], we get:
[tex]\[ \int 6x^{-3} \, dx = 6 \int x^{-3} \, dx = 6 \left( \frac{x^{-3+1}}{-3+1} \right) = 6 \left( \frac{x^{-2}}{-2} \right) = -3x^{-2} \][/tex]
[tex]\[ \int 6x^{-7} \, dx = 6 \int x^{-7} \, dx = 6 \left( \frac{x^{-7+1}}{-7+1} \right) = 6 \left( \frac{x^{-6}}{-6} \right) = -x^{-6} \][/tex]
So, the general antiderivative is:
[tex]\[ F(x) = -3x^{-2} - x^{-6} + C \][/tex]
3. Apply the initial condition:
We are given that [tex]\( F(1) = 0 \)[/tex]. Substituting [tex]\( x = 1 \)[/tex] and [tex]\( F(1) = 0 \)[/tex] into the antiderivative, we get:
[tex]\[ 0 = -3(1)^{-2} - (1)^{-6} + C \][/tex]
Simplifying, we find:
[tex]\[ 0 = -3(1) - 1 + C \implies 0 = -3 - 1 + C \implies 0 = -4 + C \implies C = 4 \][/tex]
4. Write the final solution:
Substituting the value of [tex]\( C \)[/tex] back into the antiderivative:
[tex]\[ F(x) = -3x^{-2} - x^{-6} + 4 \][/tex]
We can rewrite [tex]\( x^{-2} \)[/tex] and [tex]\( x^{-6} \)[/tex] back using positive exponents:
[tex]\[ F(x) = -\frac{3}{x^2} - \frac{1}{x^6} + 4 \][/tex]
So, the antiderivative of the function [tex]\( f(x) = \frac{6}{x^3} - \frac{6}{x^7} \)[/tex] with the initial condition [tex]\( F(1) = 0 \)[/tex] is:
[tex]\[ F(x) = -\frac{3}{x^2} - \frac{1}{x^6} + 4 \quad \text{or equivalently,} \quad F(x) = 4 - \frac{3}{x^2} - \frac{1}{x^6} \][/tex]
The function [tex]\( F(x) = 2 + \frac{1 - 3x^4}{x^6} \)[/tex] satisfies all these conditions.
1. Express the function in a more integrable form:
First, rewrite [tex]\( f(x) \)[/tex] with negative exponents:
[tex]\[ f(x) = 6x^{-3} - 6x^{-7} \][/tex]
2. Find the antiderivative:
To find the antiderivative [tex]\( F(x) \)[/tex], we integrate [tex]\( f(x) \)[/tex] term by term:
[tex]\[ F(x) = \int (6x^{-3} - 6x^{-7}) \, dx \][/tex]
Using the power rule for integration [tex]\( \int x^n \, dx = \frac{x^{n+1}}{n+1} + C \)[/tex], we get:
[tex]\[ \int 6x^{-3} \, dx = 6 \int x^{-3} \, dx = 6 \left( \frac{x^{-3+1}}{-3+1} \right) = 6 \left( \frac{x^{-2}}{-2} \right) = -3x^{-2} \][/tex]
[tex]\[ \int 6x^{-7} \, dx = 6 \int x^{-7} \, dx = 6 \left( \frac{x^{-7+1}}{-7+1} \right) = 6 \left( \frac{x^{-6}}{-6} \right) = -x^{-6} \][/tex]
So, the general antiderivative is:
[tex]\[ F(x) = -3x^{-2} - x^{-6} + C \][/tex]
3. Apply the initial condition:
We are given that [tex]\( F(1) = 0 \)[/tex]. Substituting [tex]\( x = 1 \)[/tex] and [tex]\( F(1) = 0 \)[/tex] into the antiderivative, we get:
[tex]\[ 0 = -3(1)^{-2} - (1)^{-6} + C \][/tex]
Simplifying, we find:
[tex]\[ 0 = -3(1) - 1 + C \implies 0 = -3 - 1 + C \implies 0 = -4 + C \implies C = 4 \][/tex]
4. Write the final solution:
Substituting the value of [tex]\( C \)[/tex] back into the antiderivative:
[tex]\[ F(x) = -3x^{-2} - x^{-6} + 4 \][/tex]
We can rewrite [tex]\( x^{-2} \)[/tex] and [tex]\( x^{-6} \)[/tex] back using positive exponents:
[tex]\[ F(x) = -\frac{3}{x^2} - \frac{1}{x^6} + 4 \][/tex]
So, the antiderivative of the function [tex]\( f(x) = \frac{6}{x^3} - \frac{6}{x^7} \)[/tex] with the initial condition [tex]\( F(1) = 0 \)[/tex] is:
[tex]\[ F(x) = -\frac{3}{x^2} - \frac{1}{x^6} + 4 \quad \text{or equivalently,} \quad F(x) = 4 - \frac{3}{x^2} - \frac{1}{x^6} \][/tex]
The function [tex]\( F(x) = 2 + \frac{1 - 3x^4}{x^6} \)[/tex] satisfies all these conditions.
We hope our answers were helpful. Return anytime for more information and answers to any other questions you may have. Thank you for choosing our platform. We're dedicated to providing the best answers for all your questions. Visit us again. Thank you for using Westonci.ca. Come back for more in-depth answers to all your queries.