what is the answer of integrals 1) Findthe integrals:i)i) SFInx dxdx4 -x2?REDMI NOTE 9SAI QUAD CAMERA

Answered: 1) Find the integrals: i) i) SF Inx dx…

Elementary Geometry For College Students, 7e

7th Edition

ISBN:9781337614085

Author:Alexander, Daniel C.; Koeberlein, Geralyn M.

Publisher:Alexander, Daniel C.; Koeberlein, Geralyn M.

Chapter8: Areas Of Polygons And Circles

Section8.4: Cicumference And Area Of A Cicle

Problem 18E: A rectangle has an area of 36 in2. What is the limit smallest possible value of the perimeter of the...

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Question

what is the answer of integrals

1) Find
the integrals:
i)
i) SF
Inx dx
dx
4 -x2?
REDMI NOTE 9S
AI QUAD CAMERA

Expert Solution

Step 1

$P a r t (i) \int \frac{x^{2} - 1}{\sqrt{4 - x^{2}}} d x$

Step 2

$N o w, ∴ \int \frac{x^{2} - 1}{\sqrt{4 - x^{2}}} d x = \int \frac{x^{2}}{\sqrt{4 - x^{2}}} d x - \int \frac{1}{\sqrt{4 - x^{2}}} d x L e t A = \int \frac{x^{2}}{\sqrt{4 - x^{2}}} d x A p p l y i n g t r i g o n o m e t r i c s u b s t i t u t i o n m e t h o d : x = 2 \sin (u) \Rightarrow A = \int 4 \sin^{2} (u) d \Rightarrow A = 4 \int \frac{1 - \cos (2 u)}{2} d u \Rightarrow A = 2 (\int 1 \cdot d u - \int \cos (2 u) d u) \Rightarrow A = 2 (u - \frac{1}{2} \sin (2 u)) + C \Rightarrow A = 2 (a r c \sin (\frac{1}{2} x) - \frac{1}{2} \sin (2 a r c \sin (\frac{1}{2} x))) + C \Rightarrow A = 2 a r c \sin (\frac{1}{2} x) - \sin (2 a r c \sin (\frac{1}{2} x)) + C$

Step 3

$T h e i n t e g r a l b e c o m e s = A - \int \frac{1}{\sqrt{4 - x^{2}}} d x = 2 a r c \sin (\frac{1}{2} x) - \sin (2 a r c \sin (\frac{1}{2} x)) + C - a r c \sin (\frac{1}{2} x) = a r c \sin (\frac{1}{2} x) - \sin (2 a r c \sin (\frac{1}{2} x)) + C ∴ \int \frac{x^{2} - 1}{\sqrt{4 - x^{2}}} d x = a r c \sin (\frac{1}{2} x) - \sin (2 a r c \sin (\frac{1}{2} x)) + C$

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1) Find the integrals: i) i) SF Inx dx 4-X2