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Lesson 29: Integration by Substition (worksheet)

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Lesson 29: Integration by Substition (worksheet)

  1. 1. Worksheet for Section 5.5 Integration by Substitution V63.0121, Calculus I December 7, 2009 Find the following integrals. In the case of an indefinite integral, your answer should be the most general antiderivative. In the case of a definite integral, your answer should be a number. In these problems, a substitution is given. 1. (3x − 5)17 dx, u = 3x − 5 4 2. x x2 + 9 dx, u = x2 + 9 0 √ e x √ 3. √ dx, u = x. x cos 3x dx 4. , u = 5 + 2 sin 3x 5 + 2 sin 3x In these problems, you need to determine the substitution yourself. 5. (4 − 3x)7 dx. π/3 6. csc2 (5x) dx π/4 3 −1 7. x2 e3x dx 1
  2. 2. Sometimes there is more than one way to skin a cat: x 8. Find dx, both by long division and by substituting u = 1 + x. 1+x 2z dz , both by substituting u = z 2 + 1 and u = 3 9. Find √ 3 z 2 + 1. z2 + 1 Use the trigonometric identity cos 2α = cos2 α − sin2 α = 2 cos2 α − 1 = 1 − 2 sin2 α to find 10. sin2 x dx 11. cos2 x dx ‘ 12. Find sec x dx by multiplying the numerator and denominator by sec x + tan x. 2

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