MATH 141 Exam 1

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Last updated 9:00 PM on 6/10/26
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53 Terms

1
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∫(1/x)dx =

ln|x| + C

2
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∫exdx =

ex +C

3
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∫axdx =

(ax/lna) +C

4
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∫(f’(x)/f(x))dx =

ln |f(x)| +C

5
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∫ (1/√1-x2)dx =

sin-1x +C

6
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∫ (1/√1+x2)dx =

tan-1x+C

7
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∫ sinxdx =

-cosx +C

8
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∫ cosxdx =

sinx + C

9
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∫ sec2xdx =

tanx + C

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∫ tanxdx =

ln|secx| +C

11
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∫ csc2xdx =

-cotx +C

12
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∫ secx tanxdx =

secx + C

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∫ cscx cotxdx =

-csc x +C

14
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∫ sinaxdx =

-(1/a)cosax + C

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∫ cosaxdx =

(1/a)sinx + C

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d/dx (lnx) =

1/x

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d/dx (logax) =

1/x * lna

18
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d/dx (ex) =

ex

19
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d/dx (ax) =

ax * lna

20
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∫ (1/x*lna) dx =

logax +C

21
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∫ (cosx)/(sinx)dx =

ln|sinx| + C

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ln (e) =

1

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ln (1) =

0

24
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cos (0) =

1

25
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cos (π) =

-1

26
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xndx =

(xn+1/n+1) + C when x doesn’t = -1

27
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∫ sinhxdx =

coshx + C

28
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∫ 1/(x2+a2)dx =

1/a tan-1(x/a) +C

29
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∫ cos hx dx =

sinhx + C

30
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∫ cotxdx =

ln |sinx| + C

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∫ 1/ (√a2-x2)dx =

sin-1 (x/a) + C

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by parts formula

u * v - ∫ v * du

33
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d/dx logax =

(lnx/lna)dx

34
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sin2x + cos2x = (pythagorean identity)

1

35
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sin2x = (pythagorean identity)

1 - cos2x

36
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cos2x = (pythagorean identity)

1 - sin2x

37
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sin2x = (double angle formula)

2sinxcosx

38
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cos2x = (double angle formula)

  1. cos2x - sin2x

  2. 1 - 2sin2x

  3. 2cos2x - 1

39
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sin2x = (half-angle formula)

(1-cos2x)/2

40
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cos2x = (half-angle formula)

(1+cos2x)/2

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∫sinaxdx =

(-cosax)/a + C

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∫cosaxdx

(sinax) / a + C

43
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∫sinnxcosmxdx ** if both exponents are even use:

half-angle formula

44
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∫sinnxcosmxdx if both exponents are odd (or one is even, one odd):

  1. u = sinx ; du = cosxdx

  2. u = cosx ; du = -sinxdx

45
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∫tannxsecmxdx **if both exponents are even:

  1. d/dx tanx = sec2x

  2. sec2x = 1 + tan2x

  3. let u = tanx ; du = sec2xdx

46
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∫tannxsecmxdx **if both exponents are odd:

  1. d/dx secx = secxtanx

  2. tan2x = sec2x -1

  3. let u = secx ; du = secxtanxdx

47
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trigonometric substitution table

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48
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Proper Rational Function

degree of demoninator > degree of numerator

49
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Improper Rational Function

Degree of numerator > degree of demoninator

50
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Proper Functions Procedure:

Step one: factor demoninator completely

Step two: express function as:
(A1/a1x+b1) + (A2/a2x+b2) + … + (Ak/akx+bk)

51
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if the demoninator is repeated (x2(x-1)3) for example, then:

(A1/a1x+b1) + (A2/(a1x+b1)2) + … + (Ar/(a1x+b1)r)

52
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if the denominator has an irreducible quadratic factor; (x-2)(x2+1)(x2+4), for example, then:

(Ax+B) / (ax2+bx+c)

53
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For improper functions:

∫Q(x) + (R(x)/original demoninator)

  1. Q(x) = quotient and R(x) = remainder; long divide to get these variables, then integrate.