Math 1A03 Fall 2011 Practice Midterm 2. Duration: 90 minutes. 1) Find the
following integrals. No partial credit will be given on this question, so do not
forget.
Math 1A03 Fall 2011 Practice Midterm 2 Duration: 90 minutes 1) Find the following integrals. No partial credit will be given on this question, so do not forget the arbitrary constant of integration! Z (x − 1)2 a) dx x4
Z b)
e3x+4 dx
Z c)
sin(x) cos(x) dx
Z d)
1 dx (3 − x)2
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McMaster University Math 1A03 Fall 2011 Practice Midterm 2 2) Find the following definite integrals. Z 1 (a) x2 e−4x dx 0
Z
5
(b)
10
sin(x) cos(ex + 37x6 ) dx
−5
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McMaster University Math 1A03 Fall 2011 Practice Midterm 2 3) Find the following indefinite integrals. Z (a) arctan(2x) dx
Z (b)
sin8 (x) cos3 (x) dx
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McMaster University Math 1A03 Fall 2011 Practice Midterm 2 Z √ 3 + 2x − x2 4) Find the indefinite integral dx. The final answer should not contain any x−1 trigonometric or inverse trigonometric functions.
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McMaster University Math 1A03 Fall 2011 Practice Midterm 2 Z 5) Find the indefinite integral
2x3 + 16x + 7 dx. x3 + 7x
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McMaster University Math 1A03 Fall 2011 Practice Midterm 2 6) For each of the numerical integration methods (trapezoidal, midpoint, Simpson’s) R 3 determine 1 how large n should be so that the approximations Tn , Mn and Sn to the integral 1 e x dx are accurate to within 10−3 . 1 The second derivative of the function f (x) = e x is equal to 1 2 1 00 f (x) = + 4 ex 3 x x and the fourth derivative is 1 24 36 12 1 (4) f (x) = + 6 + 7 + 8 ex . 5 x x x x
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McMaster University Math 1A03 Fall 2011 Practice Midterm 2 Derivatives (xn )0 = nxn−1 , n 6= 0
(a)0 = 0
(ex )0 = ex
(ax )0 = ax ln a
(ln x)0 =
1 x
(loga x)0 =
1 x ln a
(sin x)0 = cos x
(cos x)0 = − sin x
(tan x)0 = sec2 x
(cot x)0 = − csc2 x
(sec x)0 = sec x tan x
(csc x)0 = − csc x cot x
(arcsin x)0 = √ (arctan x)0 =
1 1 − x2
(arccos x)0 = − √
1 1 + x2
(arccot x)0 = −
1 1 − x2
1 1 + x2
Integrals (constants of integration are omitted) Z
Z
xn+1 , n 6= −1 x dx = n+1 n
x
x
Z
Z
e dx = e Z
Z
ax ln a
cos x dx = sin x Z
tan x dx = − ln | cos x | Z
cot x dx = ln | sin x | Z
sec x dx = ln | sec x + tan x | 2
csc x dx = − ln | csc x + cot x | Z
sec x dx = tan x Z
csc2 x dx = − cot x
Z csc x cot x dx = − csc x
sec x tan x dx = sec x Z
ax dx =
Z sin x dx = − cos x
Z
1 dx = ln |x | x
x 1 1 dx = arctan x 2 + a2 a a
Z
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√
x 1 dx = arcsin a a2 − x2
McMaster University Math 1A03 Fall 2011 Practice Midterm 2 Trigonometry sec x =
