Use integration by parts to obtain the formuladr.sarating Invorcof Funotiono

Given:∫1-x2 dxIntegration formula:∫sinx=-cosx+C∫xn dx=xn+1n+1

Answered: Use integration by parts to obtain the…

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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Question

Use integration by parts to obtain the formula
dr.
sarating Invorc
of Funotiono

Expert Solution

Step 1 Let us evaluate the given integral

$G i v e n : \int \sqrt{1 - x^{2}} d x I n t e g r a t i o n f o r m u l a : \int \sin x = - \cos x + C \int x^{n} d x = \frac{x^{n + 1}}{n + 1}$

Step 2

$\int \sqrt{1 - x^{2}} d x x = \sin u d x = \cos u d u = \int \sqrt{1 - \sin^{2} u} \cos u d u = \int \sqrt{\cos^{2} u} \cos u d u = \int \cos u . \cos u d u = \int \cos^{2} u d u = \frac{1}{2} \int 1 + \cos (2 u) d u [\cos^{2} x = \frac{1}{2} (1 + \cos 2 x)] = \frac{1}{2} \int d u + \frac{1}{2} \int \cos (2 u) d u = \frac{1}{2} u + \frac{1}{2} \frac{\sin 2 u}{2} = \frac{1}{2} u + \frac{1}{2} \times \frac{2 \sin u \cos u}{2} = \frac{1}{2} u + \frac{\sin u \cos u}{2} x = \sin u \cos u = \sqrt{1 - \sin^{2} u} \cos u = \sqrt{1 - x^{2}} = \frac{1}{2} s i n^{- 1} (x) + \frac{1}{2} x \sqrt{1 - x^{2}}$