problemset10s - MAT 137Y 2008-09 Winter Session Solutions...

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 2008-09 Winter Session, Solutions to Problem Set 10 1 (SHE 8.2) 6. We integrate by parts and let u = x 2 and dv = xe- x 2 dx . Then du = 2 x dx and v =- 1 2 e- x 2 . Therefore Z x 3 e- x 2 dx =- x 2 2 e- x 2 + Z xe- x 2 dx =- 1 2 x 2 e- x 2- 1 2 e- x 2 + C . 20. We integrate by parts twice. First we let u = x 2 and dv = sin x dx . Then du = 2 x dx and v =- cos x . Hence, Z π / 2 x 2 sin x dx = h- x 2 cos x i π / 2 + Z π / 2 2 x cos x dx = h- x 2 cos x i π / 2 + h 2 x sin x i π / 2- Z π / 2 2sin x dx = h- x 2 cos x + 2 x sin x + 2cos x i π / 2 = π- 2 . 38. Let u = cos ( ln x ) , dv = dx , du =- sin ( ln x ) x dx , v = x . Then Z cos ( ln x ) dx = x cos ( ln x )+ Z sin ( ln x ) dx = x cos ( ln x )+ x sin ( ln x )- Z cos ( ln x ) dx ( u = sin ( ln x ) , dv = dx , du = cos ( ln x ) x dx , v = x ) Therefore, Z cos ( ln x ) dx = 1 2 x [ cos ( ln x )+ sin ( ln x )]+ C . 2 (SHE 8.3) 6. Z sin 3 x cos 2 x dx = Z cos 2 x ( 1- cos 2 x ) sin x dx =- 1 3 cos 3 x + 1 5 cos 5 x + C . 20. Z π / 2 cos 4 x dx = Z π / 2 1 + cos2 x 2 2 dx = Z π / 2 1 4 + 1 2 cos2 x + cos 2 2 x 4 dx = Z π / 2 1 4 + 1 2 cos2 x + 1 + cos4 x 8 dx = 3 8 x + 1 4 sin2 x + 1 32 sin4 x π / 2 = 3 π 16 . 28. Z sec 4 3 xdx = Z ( 1 + tan 2 3 x ) sec 2 3 xdx = 1 9 tan 3 3 x + 1 3 tan3 x + C . 3 (SHE 8.4) 8. Let x = 2tan u = ⇒ dx = 2sec 2 udu . Thus Z x 2 √ 4 + x 2 dx = Z 8tan 2 u sec 2 u p 4 ( 1 + tan 2 u ) du = 4 Z tan 2 u sec udu = 4 Z ( sec 3 u- sec u ) du = 4 1 2 sec u tan u- 1 2 ln | sec u + tan u | + C , where the integral of sec 3 x is solved in Section 8.3 Example 9.is solved in Section 8....
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.

Page1 / 5

problemset10s - MAT 137Y 2008-09 Winter Session Solutions...

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