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# MATH 1226 Unit 1 Practice 1 (Basic Integration Techniques) MATH 1226

Virginia Tech

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This 2 page Bundle was uploaded by Luke Notetaker on Saturday September 3, 2016. The Bundle belongs to MATH 1226 at Virginia Polytechnic Institute and State University taught by in Fall 2016. Since its upload, it has received 20 views. For similar materials see Calculus of a Single Variable in Math at Virginia Polytechnic Institute and State University.

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Date Created: 09/03/16

MATH 1226 Practice 1 Z 2 1. (2x ▯ 3x + 5)dx Z 2. (x + 1) dx Z 1 3. dx (where a is a constant) x ▯ a Z ▯ 4. 23dx (x ▯ a) Z 5. 1 dx (where k is a constant and k > 1) (x ▯ a)k Z lnjxj 6. dx x Z 7. sinmxdx (where m is a constant) Z 8. cos(mx + n)dx (where m and n are constants) Z 72 9. e▯3x+e dx Z p 4 10. (1 + x) dx Z 2 (x + 1)(x ▯ 3) 11. 3x2 dx Z 12. p 1 dx 1 ▯ x2 Z 1 13. p dx (where c is a constant) c ▯ x 2 Z 1 14. dx (x + a)(x + b) Z 1 15. (x ▯ 5x + 6 dx Z 1 16. dx 4x + 4x ▯ 3 Z (e + 1)(e2x + 3) 17. dx ex Z ax + b 18. dx (where a,b,c and d are constants) cx + d 1 MATH 1226 Practice 1 Z Z Z Z 2 2 2 3 3 2 1. (2x ▯ 3x + 5)dx = 2x dx ▯ 3xdx + 5dx = x ▯ x + 5x + C 3 2 Z Z Z Z Z 1 3 2. (x + 1) dx = x dx + 3x dx + 3x dx + dx = x + x + x + x + C 7 5 Z Z 1 1 3. dx = d(x ▯ a) = lnjx ▯ aj + C x ▯ a x ▯ a Z Z 4. ▯ dx = ▯ (x ▯ a)23d(x ▯ a) = ▯ ▯ (x ▯ a)22 + C = ▯ ▯ + C (x ▯ a)3 22 22(x ▯ a)2 Z Z 1 ▯k 1 ▯k+1 1 5. kdx = (x ▯ a) d(x ▯ a) = (x ▯ a) + C = ▯ k▯1 + C (x ▯ a) ▯k + 1 (k ▯ 1)(x ▯ a) Z Z Z lnjxj 1 1 2 6. x dx = lnjxjx dx) = lnjxjd(lnjxj) 2 ln jxj + C Z 1 Z 1 Z 1 7. sinmxdx = sinmx(mdx) = (sinmx)d(mx) = ▯ cosmx + C m m m Z Z Z 1 1 1 8. cos(mx + n)dx = m cos(mx + n)(mdx) = m cos(mx + n)d(mx + n) = m sin(mx + n) + C Z 72 1 Z 72 1Z 72 1 72 9. e3x+e dx = ▯ e▯3x+e [(▯3)dx] = ▯ e▯3x+e d(▯3x + e ) = ▯ e▯3x+e + C 3 3 3 Z Z p 4 2 3 p 1 3 12 5 2 8 3 10. (1 + x) dx = (x + 4x 2+ 6x + 4 x + 1)dx = x + x3 + 3x + x2 + x + C 3 5 3 Z 2 Z 11. (x + 1)(x ▯ 3)dx = ( x + 1 ▯ 1 ▯ 1 )dx = 1x + 1x ▯ lnjxj +1 + C 3x2 3 3 x x 2 6 3 x Z 1 12. p 2dx = arcsinx + C 1 ▯ x Z Z Z p 1 1 p 1 p 1 x 13. c ▯ x 2dx = c 1 ▯ (x=c)dx = 1 ▯ (x=c)d(c) = arcsin(x=c) + C Z ▯ ▯ Z ▯ ▯ ▯ ▯ ▯ ▯ 14. 1 dx = 1 1 ▯ 1 dx = 1 ln▯x + b▯+ C (x + a)(x + b) a ▯ b x + b x + a a ▯ b ▯x + a▯ ▯ ▯ Z 1 Z ▯ 1 ▯▯ 1 ▯ Z ▯ 1 1 ▯ ▯x ▯ 3▯ 15. dx = dx = ▯ dx = ln▯ ▯+ C (x2▯ 5x + 6) x ▯ 3 x ▯ 2 x ▯ 3 x ▯ 2 ▯x ▯ 2▯ Z Z ▯ ▯▯ ▯ Z ▯ ▯ ▯ ▯ ▯ ▯ 1 1 1 1 1 1 1 ▯2x ▯ 3▯ 16. 2 dx = dx = ▯ dx = ln▯ ▯+ C 4x + 4x ▯ 3 2x ▯ 3 2x + 1 4 2x ▯ 3 2x + 1 4 2x + 1 Z x 2x Z (e + 1)(e + 3) 2x x ▯x 1 2x x ▯x 17. ex dx = (e + e + 3 + 3e )dx = 2e + e + 3x ▯ 3e + C Z Z ▯ ▯ ▯ ▯ 18. ax + bdx = a + b ▯ ad=c dx = a x + bc ▯ ad lnjcx + dj + C cx + d c cx + d c c2 2

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