spring03 MTsolns

Thomas' Calculus: Early Transcendentals

Info iconThis preview shows pages 1–3. Sign up to view the full content.

View Full Document Right Arrow Icon

Info iconThis preview has intentionally blurred sections. Sign up to view the full version.

View Full Document Right Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MAT 104 Midterm March 2003 Problem Possible score Average score 1 12 9.5 2 12 4.9 3 12 8.9 4 12 5.4 5 12 7.5 6 12 8.1 7 12 6.8 8 6 4.8 9 10 4.3 Total 100 60.2 1. (12 points) Find Z sin 3 (ln x ) cos 2 (ln x ) x dx . Let u = ln x . Then the integral becomes Z sin 3 u cos 2 u du = Z sin 2 u cos 2 u sin u du = Z (1- cos 2 u ) cos 2 u sin u du. Substitute t = cos u to get- Z (1- t 2 ) t 2 dt = Z ( t 4- t 2 ) dt = t 5 5- t 3 3 + C = cos 5 (ln x ) 5- cos 3 (ln x ) 3 + C. 2. (12 points) Find Z ln( x 2 + x + 1) x 2 dx . Do integration by parts with u = ln( x 2 + x + 1) and dv = dx/x 2 . Then du = (2 x + 1) dx/ ( x 2 + x + 1) and v =- 1 /x . So Z ln( x 2 + x + 1) x 2 dx =- 1 x ln( x 2 + x + 1) + Z 2 x + 1 x ( x 2 + x + 1) Now use partial fractions to compute the integral on the right. 2 x + 1 x ( x 2 + x + 1) = A x + Bx + C x 2 + x + 1 = Ax 2 + Ax + A + Bx 2 + Cx x ( x 2 + x + 1) This leads to 0 = A + B , 2 = A + C , 1 = A and we conclude that B =- 1 and C = 1. So now we have Z 2 x + 1 x ( x 2 + x + 1) dx = Z dx x + Z 1- x x 2 + x + 1 dx = ln | x | + Z 1- x ( x + 1 / 2) 2 + 3 / 4 dx. In this integral we make the substitution u = x + 1 / 2 and then- u + 3 / 2 =- x + 1. So we get ln | x | + Z 1- x ( x + 1 / 2) 2 + 3 / 4 dx = ln | x | - 1 2 Z 2 u u 2 + 3 / 4 du + 3 2 Z du u 2 + 3 / 4 = ln | x | - 1 2 ln( u 2 + 3 / 4) + 2 √ 3 3 2 arctan 2 u √ 3 ! + C Returning to the original variable x and combining all the pieces gives- 1 x ln( x 2 + x + 1) + ln | x | - 1 2 ln( x 2 + x + 1) + √ 3 arctan 2 x + 1 √ 3 !...
View Full Document

{[ snackBarMessage ]}

Page1 / 5

spring03 MTsolns - MAT 104 Midterm March 2003 Problem...

This preview shows document pages 1 - 3. Sign up to view the full document.

View Full Document Right Arrow Icon
Ask a homework question - tutors are online