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 CalculusIn 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

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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