# Definite, indefinite integrals, and improper integrals.

May 19th, 2015
Studypool Tutor
Price: \$10 USD

Tutor description

how to use partial fractions, substitution and finding the arc length. Also examples and explanations on undetermined forms in Calculus are found in this document

Word Count: 1002
Showing Page: 1/5
Mathematics 112 Spring 11: Solutions to homework #1Definite and indefinite integralsZi)2xdx. Using partial fraction,(x 1)(x2 + 1)2xABx + C=+ 2,2(x 1)(x + 1)x1x +1which leads to A = C = 1, B = 1. Hence,ZZZ2x1x + 1dx =dx +dx2(x 1)(x + 1)x1x2 + 1ZZZx11dx dx +dx=22x1x +1x +11= ln |x 1| ln |x2 + 1| + arctan x + C2Z2xii)dx. Again, using partial fraction,(x 1)(x + 1)2ABC2x=++,2(x 1)(x + 1)x 1 x + 1 (x + 1)2which leads to A = 12 , B = 21 , C = 1. Hence,ZZZZZ2x11111dx =dx dx +dx2(x 1)(x + 1)2x12x+1(x + 1)2111= ln |x 1| ln |x + 1| 22x+1/2iii)sin3 (x) cos3 (x)dx. We want to use substitution./6Z/23/2Z3sin3 (x) cos2 (x) cos(x)(x)dxsin (x) cos (x)dx =/6/6/2Zsin3 (x)(1 sin2 (x)) cos(x)(x)dx=/61Z321Z(u3 u5 )duu (1 u )du =u = sin(x), du = cos(x)dx =1212u4 u6=469=128 1 11111 =4 664 384212Z /2iv)x3 sin(x2 )dx. We need to use both substitution and integration by parts. First let0t = x2 , dt = 2xdx, so thatZ /20/2Z1x sin(x )dx =232t sin(t)dt0Then we integrate by parts:((u=tdu = dtdv = sin(t)dtv = cos(t)Z /2Z/2 1 /211t sin(t)dt = t cos(t) +cos(t)dt(remember that cos(/2) = 0)2 022 00/2 11= sin(t)

## Review from student

Studypool Student
" Thanks, good work "

1821 tutors are online

Brown University

1271 Tutors

California Institute of Technology

2131 Tutors

Carnegie Mellon University

982 Tutors

Columbia University

1256 Tutors

Dartmouth University

2113 Tutors

Emory University

2279 Tutors

Harvard University

599 Tutors

Massachusetts Institute of Technology

2319 Tutors

New York University

1645 Tutors

Notre Dam University

1911 Tutors

Oklahoma University

2122 Tutors

Pennsylvania State University

932 Tutors

Princeton University

1211 Tutors

Stanford University

983 Tutors

University of California

1282 Tutors

Oxford University

123 Tutors

Yale University

2325 Tutors