sol3-06 - MAT 137Y 2006-2007 Winter Session, Solutions to...

Info iconThis preview shows pages 1–2. 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 DocumentRight Arrow Icon
This is the end of the preview. Sign up to access the rest of the document.

Unformatted text preview: MAT 137Y 2006-2007 Winter Session, Solutions to Term Test 3 1. Evaluate the following integrals. (8%) (i) Z sec 3 x tan x dx . Z sec 3 x tan xdx = Z sec 2 x sec x tan xdx = 1 3 sec 3 x + C (by a simple substitution u = sec x ). (8%) (ii) Z 1 x 3 x dx . Integrating by parts, let u = x , dv = 3 x dx . Then du = dx and v = 3 x ln3 . Hence Z 1 x 3 x dx = x 3 x ln3 1- Z 1 3 x ln3 dx = x 3 x ln3- 3 x ( ln3 ) 2 1 = 3 ln3- 3 ( ln3 ) 2- + 1 ( ln3 ) 2 = 3 ln3- 2 ( ln3 ) 2 = 3ln3- 2 ( ln3 ) 2 . (10%) (iii) Z dx x 4 x 2- 1 . Let x = sec . Then dx = sec tan d , thus we have Z sec tan sec 4 sec 2 - 1 d = Z tan sec 3 tan d = Z d sec 3 = Z cos 3 d . To integrate cos 3 we take out one power of cos and apply a substitution: Z cos 3 d = Z ( 1- sin 2 ) cos d = Z cos - sin 2 cos d = sin - 1 3 sin 3 + C . Substituting back, we can use triangles to see that if sec = x 1 , then sin = ( x 2- 1 ) / x . Hence Z dx x 4 x 2- 1 = x 2- 1 x- 1 3 ( x 2- 1 ) 3 / 2 x 3 ! + C . (10%) (iv) Z dx x 3 ( x + 1 ) 2 . We decompose the integrand using partial fractions: 1 x 3 ( x + 1 ) 2 = A x + B x 2 + C x 3 + D x + 1 + E ( x + 1 ) 2 Obtaining a common denominator on the right side (and ignoring the denominator) gives 1 = Ax 2 ( x + 1 ) 2 + Bx ( x + 1 ) 2 + C ( x + 1 ) 2 + Dx 3 ( x + 1 )+ Ex 3 = A ( x 4 + 2 x 3 + x 2 )+ B ( x 3 + 2 x 2 + x...
View Full Document

This note was uploaded on 04/09/2010 for the course MAT 137 taught by Professor Uppal during the Spring '08 term at University of Toronto- Toronto.

Page1 / 4

sol3-06 - MAT 137Y 2006-2007 Winter Session, Solutions to...

This preview shows document pages 1 - 2. Sign up to view the full document.

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