2. Use the Fundamental Theorem of Calculus, (a) (b) (c) to evaluate the integrals: TT 3 sec x (secx + tan x) dx 0 11/3 50Th² secon ( See (in) + tance)) ok: 5²2 ( Sic² (17) + Selled kances aux = = [² ³ Sec col de + 1²³3 (seclis ton Max [sic²(x) dx = tante, (scoa ton²x) dx = => [tan (and ] == + [sec (x)}]} 0 = [tan () - tan/] + [See (5) - secilo)) = [13-0) +(2+] [*(4x5/2 + (4x5/2+3x3/2 - 2x¹/²) dx Cπ/6 [ (127 - 12/31 - cos x) dx x3 -3π/2 scelite) • See (x) ( =
2. Use the Fundamental Theorem of Calculus, (a) (b) (c) to evaluate the integrals: TT 3 sec x (secx + tan x) dx 0 11/3 50Th² secon ( See (in) + tance)) ok: 5²2 ( Sic² (17) + Selled kances aux = = [² ³ Sec col de + 1²³3 (seclis ton Max [sic²(x) dx = tante, (scoa ton²x) dx = => [tan (and ] == + [sec (x)}]} 0 = [tan () - tan/] + [See (5) - secilo)) = [13-0) +(2+] [*(4x5/2 + (4x5/2+3x3/2 - 2x¹/²) dx Cπ/6 [ (127 - 12/31 - cos x) dx x3 -3π/2 scelite) • See (x) ( =
Calculus: Early Transcendentals
8th Edition
ISBN:9781285741550
Author:James Stewart
Publisher:James Stewart
Chapter1: Functions And Models
Section: Chapter Questions
Problem 1RCC: (a) What is a function? What are its domain and range? (b) What is the graph of a function? (c) How...
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Answer b and c. Also pls write out the answer so it will be easier to understand. Thank you!
![## Evaluating Integrals Using the Fundamental Theorem of Calculus
In this exercise, we will use the Fundamental Theorem of Calculus to evaluate the given integrals.
### Problem 2
#### (a)
Evaluate the integral:
\[
\int_{0}^{\pi/3} \sec x (\sec x + \tan x) \, dx
\]
**Solution Steps:**
1. Expand the integrand:
\[
\sec x (\sec x + \tan x) = \sec^2 x + \sec x \tan x
\]
2. Integrate term by term:
\[
\int_{0}^{\pi/3} \sec^2 x \, dx + \int_{0}^{\pi/3} \sec x \tan x \, dx
\]
3. Evaluate each integral:
- \(\int \sec^2 x \, dx = \tan x\)
- \(\int \sec x \tan x \, dx = \sec x\)
4. Apply the limits:
\[
\left[ \tan x \right]_0^{\pi/3} + \left[ \sec x \right]_0^{\pi/3} = \left( \sqrt{3} - 0 \right) + \left(2 - 1\right) = \sqrt{3} + 1
\]
The handwritten note confirms this result:
\[
\int_{0}^{\pi/3} \sec x (\sec x + \tan x) \, dx = \sqrt{3} + 1
\]
#### (b)
Evaluate the integral:
\[
\int_{1}^{4} (4x^{5/2} + 3x^{3/2} - 2x^{1/2}) \, dx
\]
Evaluate each term using the antiderivative of \(x^n\), which is \(\frac{x^{n+1}}{n+1}\).
#### (c)
Evaluate the integral:
\[
\int_{-3\pi/2}^{\pi/6} \left( \frac{1}{x^2} - \frac{1}{x^3} - \cos x \right) \, dx
\]
Evaluate each term separately using standard integration techniques](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F7866e7eb-0860-4fe3-a693-6d7d638dcf8c%2F54f01c01-5e6a-4a18-a09f-229591407cf0%2Fjpvdf3q_processed.png&w=3840&q=75)
Transcribed Image Text:## Evaluating Integrals Using the Fundamental Theorem of Calculus
In this exercise, we will use the Fundamental Theorem of Calculus to evaluate the given integrals.
### Problem 2
#### (a)
Evaluate the integral:
\[
\int_{0}^{\pi/3} \sec x (\sec x + \tan x) \, dx
\]
**Solution Steps:**
1. Expand the integrand:
\[
\sec x (\sec x + \tan x) = \sec^2 x + \sec x \tan x
\]
2. Integrate term by term:
\[
\int_{0}^{\pi/3} \sec^2 x \, dx + \int_{0}^{\pi/3} \sec x \tan x \, dx
\]
3. Evaluate each integral:
- \(\int \sec^2 x \, dx = \tan x\)
- \(\int \sec x \tan x \, dx = \sec x\)
4. Apply the limits:
\[
\left[ \tan x \right]_0^{\pi/3} + \left[ \sec x \right]_0^{\pi/3} = \left( \sqrt{3} - 0 \right) + \left(2 - 1\right) = \sqrt{3} + 1
\]
The handwritten note confirms this result:
\[
\int_{0}^{\pi/3} \sec x (\sec x + \tan x) \, dx = \sqrt{3} + 1
\]
#### (b)
Evaluate the integral:
\[
\int_{1}^{4} (4x^{5/2} + 3x^{3/2} - 2x^{1/2}) \, dx
\]
Evaluate each term using the antiderivative of \(x^n\), which is \(\frac{x^{n+1}}{n+1}\).
#### (c)
Evaluate the integral:
\[
\int_{-3\pi/2}^{\pi/6} \left( \frac{1}{x^2} - \frac{1}{x^3} - \cos x \right) \, dx
\]
Evaluate each term separately using standard integration techniques
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