Westonci.ca is your go-to source for answers, with a community ready to provide accurate and timely information. Our Q&A platform offers a seamless experience for finding reliable answers from experts in various disciplines. Our platform offers a seamless experience for finding reliable answers from a network of knowledgeable professionals.
Sagot :
Let's proceed with evaluating the given integrals step-by-step based on the provided information:
### Given:
1. [tex]\(\int_1^3 f(x) \, dx = 5\)[/tex]
2. [tex]\(\int_3^8 f(x) \, dx = -4\)[/tex]
3. [tex]\(\int_3^8 g(x) \, dx = 6\)[/tex]
### Part a: [tex]\(\int_1^3 6 f(x) \, dx\)[/tex]
To find [tex]\(\int_1^3 6 f(x) \, dx\)[/tex], we use the property of integrals which states that multiplying the integrand by a constant can be factored out:
[tex]\[ \int_1^3 6 f(x) \, dx = 6 \cdot \int_1^3 f(x) \, dx \][/tex]
Substitute the given value for [tex]\(\int_1^3 f(x) \, dx\)[/tex]:
[tex]\[ \int_1^3 6 f(x) \, dx = 6 \cdot 5 = 30 \][/tex]
### Part b: [tex]\(\int_3^8 -9 g(x) \, dx\)[/tex]
For [tex]\(\int_3^8 -9 g(x) \, dx\)[/tex], again, we use the property of integrals involving a constant multiple:
[tex]\[ \int_3^8 -9 g(x) \, dx = -9 \cdot \int_3^8 g(x) \, dx \][/tex]
Substitute the given value for [tex]\(\int_3^8 g(x) \, dx\)[/tex]:
[tex]\[ \int_3^8 -9 g(x) \, dx = -9 \cdot 6 = -54 \][/tex]
### Part c: [tex]\(\int_3^8 [9 f(x) - g(x)] \, dx\)[/tex]
To evaluate [tex]\(\int_3^8 [9 f(x) - g(x)] \, dx\)[/tex], we can use the linearity property of integrals:
[tex]\[ \int_3^8 [9 f(x) - g(x)] \, dx = \int_3^8 9 f(x) \, dx - \int_3^8 g(x) \, dx \][/tex]
We evaluate each integral separately:
[tex]\[ \int_3^8 9 f(x) \, dx = 9 \cdot \int_3^8 f(x) \, dx \][/tex]
[tex]\[ \int_3^8 g(x) \, dx = \int_3^8 g(x) \, dx \][/tex]
Now, substitute the given values:
[tex]\[ \int_3^8 9 f(x) \, dx = 9 \cdot (-4) = -36 \][/tex]
[tex]\[ \int_3^8 g(x) \, dx = 6 \][/tex]
Combine them:
[tex]\[ \int_3^8 [9 f(x) - g(x)] \, dx = -36 - 6 = -42 \][/tex]
### Summary of Results:
- a. [tex]\(\int_1^3 6 f(x) \, dx = 30\)[/tex]
- b. [tex]\(\int_3^8 -9 g(x) \, dx = -54\)[/tex]
- c. [tex]\(\int_3^8 [9 f(x) - g(x)] \, dx = -42\)[/tex]
Final answer for part c:
[tex]\[ \int_3^8 [9 f(x) - g(x)] \, dx = -42 \][/tex]
### Given:
1. [tex]\(\int_1^3 f(x) \, dx = 5\)[/tex]
2. [tex]\(\int_3^8 f(x) \, dx = -4\)[/tex]
3. [tex]\(\int_3^8 g(x) \, dx = 6\)[/tex]
### Part a: [tex]\(\int_1^3 6 f(x) \, dx\)[/tex]
To find [tex]\(\int_1^3 6 f(x) \, dx\)[/tex], we use the property of integrals which states that multiplying the integrand by a constant can be factored out:
[tex]\[ \int_1^3 6 f(x) \, dx = 6 \cdot \int_1^3 f(x) \, dx \][/tex]
Substitute the given value for [tex]\(\int_1^3 f(x) \, dx\)[/tex]:
[tex]\[ \int_1^3 6 f(x) \, dx = 6 \cdot 5 = 30 \][/tex]
### Part b: [tex]\(\int_3^8 -9 g(x) \, dx\)[/tex]
For [tex]\(\int_3^8 -9 g(x) \, dx\)[/tex], again, we use the property of integrals involving a constant multiple:
[tex]\[ \int_3^8 -9 g(x) \, dx = -9 \cdot \int_3^8 g(x) \, dx \][/tex]
Substitute the given value for [tex]\(\int_3^8 g(x) \, dx\)[/tex]:
[tex]\[ \int_3^8 -9 g(x) \, dx = -9 \cdot 6 = -54 \][/tex]
### Part c: [tex]\(\int_3^8 [9 f(x) - g(x)] \, dx\)[/tex]
To evaluate [tex]\(\int_3^8 [9 f(x) - g(x)] \, dx\)[/tex], we can use the linearity property of integrals:
[tex]\[ \int_3^8 [9 f(x) - g(x)] \, dx = \int_3^8 9 f(x) \, dx - \int_3^8 g(x) \, dx \][/tex]
We evaluate each integral separately:
[tex]\[ \int_3^8 9 f(x) \, dx = 9 \cdot \int_3^8 f(x) \, dx \][/tex]
[tex]\[ \int_3^8 g(x) \, dx = \int_3^8 g(x) \, dx \][/tex]
Now, substitute the given values:
[tex]\[ \int_3^8 9 f(x) \, dx = 9 \cdot (-4) = -36 \][/tex]
[tex]\[ \int_3^8 g(x) \, dx = 6 \][/tex]
Combine them:
[tex]\[ \int_3^8 [9 f(x) - g(x)] \, dx = -36 - 6 = -42 \][/tex]
### Summary of Results:
- a. [tex]\(\int_1^3 6 f(x) \, dx = 30\)[/tex]
- b. [tex]\(\int_3^8 -9 g(x) \, dx = -54\)[/tex]
- c. [tex]\(\int_3^8 [9 f(x) - g(x)] \, dx = -42\)[/tex]
Final answer for part c:
[tex]\[ \int_3^8 [9 f(x) - g(x)] \, dx = -42 \][/tex]
We hope our answers were useful. Return anytime for more information and answers to any other questions you have. Thanks for stopping by. We strive to provide the best answers for all your questions. See you again soon. Your questions are important to us at Westonci.ca. Visit again for expert answers and reliable information.