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Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347

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BuyFindarrow_forward

Calculus (MindTap Course List)

11th Edition
Ron Larson + 1 other
ISBN: 9781337275347
Textbook Problem

Deriving a Sum Derive Euler’s famous result that was mentioned in Section 9.3, n = 1 1 n 2 = π 2 6 by completing each step.

(a) Prove that d v 2 u 2 + v 2 = 1 2 u 2 arctan v 2 u 2 + C

(b) Prove that I 1 = 0 2 / 2 u u 2 2 u 2 + v 2 d v d u = π 2 18 by using the substitution u = 2 sin θ .

(c) Prove that I 2 = 2 / 2 2 u 2 u + 2 2 2 u 2 + v 2 d v d u = 4 π / 6 π / 2 arctan 1 sin θ cos θ d θ by using the substitution u = 2 sin θ .

(d) Prove the trigonometric identity

1 sin θ cos θ = tan [ ( π / 2 ) θ 2 ] .

(e) Prove that I 2 = 2 / 2 2 u 2 u + 2 2 2 u 2 + v 2 d v d u = π 2 9 .

(f) Use the formula for the sum of an infinite geometric series to verify that

n = 1 1 n 2 = 0 1 0 1 1 1 x y d x d y .

(g) Use the change of variables

u = x + y 2 and v = y x 2 to prove that

n = 1 1 n 2 = 0 1 0 1 1 1 x y d x d y = I 1 + I 2 = π 2 6 .

(a)

To determine

To prove: The given equation dv2u2+v2=12u2arctan(v2u2)+c.

Explanation

Given:

The equation provided is,

dv2u2+v2=12u2arctan(v2u2)+c

Formula used:

Integration formula,

dxa2+x2=1aarctan(xa)+c

Proof:

Take the provided equation into consideration.

dv2u2+v2

The above equation can be rewritten as,

(b)

To determine

To prove: The given equation I1=022uu22u2v2dvdu=π218 by substituting u=2sinθ.

(c)

To determine

To prove: The provided equation I2=222u2u+222u2+v2dvdu=4π6π2arctan(1sinθcosθ)dθ by substituting u=2sinθ.

(d)

To determine

To prove: The provided trigonometric equation 1sinθcosθ=tan(π2θ2).

(e)

To determine

To prove: The provided equation I2=222u2u+222u2+v2dvdu=π29.

(f)

To determine

To prove: The provided equation n=11n2=010111xydxdy.

(g)

To determine

To prove: The expression:

n=11n2=010111xydxdy=I1+I2=π26, using change of variables, u=x+y2,v=xy2.

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Chapter 14 Solutions

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P-16RECh-14 P-17RECh-14 P-18RECh-14 P-19RECh-14 P-20RECh-14 P-21RECh-14 P-22RECh-14 P-23RECh-14 P-24RECh-14 P-25RECh-14 P-26RECh-14 P-27RECh-14 P-28RECh-14 P-29RECh-14 P-30RECh-14 P-31RECh-14 P-32RECh-14 P-33RECh-14 P-34RECh-14 P-35RECh-14 P-36RECh-14 P-37RECh-14 P-38RECh-14 P-39RECh-14 P-40RECh-14 P-41RECh-14 P-42RECh-14 P-43RECh-14 P-44RECh-14 P-45RECh-14 P-46RECh-14 P-47RECh-14 P-48RECh-14 P-49RECh-14 P-50RECh-14 P-51RECh-14 P-52RECh-14 P-53RECh-14 P-54RECh-14 P-55RECh-14 P-56RECh-14 P-57RECh-14 P-59RECh-14 P-60RECh-14 P-61RECh-14 P-62RECh-14 P-63RECh-14 P-64RECh-14 P-65RECh-14 P-66RECh-14 P-67RECh-14 P-68RECh-14 P-69RECh-14 P-70RECh-14 P-71RECh-14 P-72RECh-14 P-73RECh-14 P-74RECh-14 P-75RECh-14 P-76RECh-14 P-77RECh-14 P-78RECh-14 P-1PSCh-14 P-2PSCh-14 P-3PSCh-14 P-4PSCh-14 P-5PSCh-14 P-6PSCh-14 P-7PSCh-14 P-8PSCh-14 P-9PSCh-14 P-10PSCh-14 P-11PSCh-14 P-12PSCh-14 P-14PSCh-14 P-15PSCh-14 P-16PSCh-14 P-18PS

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