m) eâx n) cosx o) sin x p) sin 3x q) cos 2x r) 5. 2. You should be able to use the table when variables other than x a
8.7
Table of Integrals Engineers usually refer to a table of integrals when performing calculations involving integration. This leaflet provides such a table. Sometimes restrictions need to be placed on the values of some of the variables. These restrictions are shown in the third column.
1. A table of integrals f (x) k, any constant x x2 xn x−1 = x1 ex ekx cos x cos kx sin x sin kx tan x sec x cosec x cot x cosh x sinh x tanh x coth x 1 x2 +a2 1 x2 −a2 1 a2 −x2 √ 1 x2 +a2 √ 1 x2 −a2 √ 1 x2 +k √ 1 a2 −x2
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R
f (x) dx
kx + c x2 +c 2 x3 +c 3 xn+1 +c n+1 ln |x| + c ex + c 1 kx e +c k sin x + c 1 sin kx + c k − cos x + c − k1 cos kx + c ln(sec x) + c ln(sec x + tan x) + c ln(cosec x−cot x) + c ln(sin x) + c sinh x + c cosh x + c ln cosh x + c ln sinh x + c 1 tan−1 xa + c a 1 ln x−a +c 2a x+a 1 a+x ln a−x + c 2a sinh−1 xa + c cosh−1 xa + c √ ln(x + x2 + k) + c sin−1 xa + c
8.7.1
n 6= −1
− π2 < x < π2 − π2 < x < π2 0 0 |x| < a a>0 x>a>0 −a 6 x 6 a
c Pearson Education Ltd 2000
Exercises 1. In each case, use the Table of Integrals to integrate the given function with respect to x. a) x b) x6 c) x−2 d) x−3 e) x−1 , (be careful!) f) x1/2 g) x−1/2 h) e3x i) e7x j) e−2x k) e0.5x l) ex m) e−x n) cos x o) sin x p) sin 3x q) cos 2x r) 5 2. You should be able to use the table when variables other than x are involved. Use the table to integrate each of the following functions with respect to t. √ a) et , b) e5t , c) t7 , d) t, e) cos 5t, f) e−t . Answers 2 1. a) x2 + c, b) − 2x12 + c, e) ln |x| + c,
f)
x7 7
x3/2 3/2
+ c,
c)
+ c = −x−1 + c, or − x1 + c,
+ c = 32 x3/2 + c, g)
j) − 21 e−2x + c,
k) 2e0.5x + c,
2. a) et + c, b)
e5t 5
p) − 13 cos 3x + c,
x−1 −1
q)
1 2
+ c,
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+ c = 2x1/2 + c,
x−2 −2
t8 8
r) 5x + c.
+ c, d)
2t3/2 3
+ c, e)
8.7.2
sin 5t 5
+ c = − 12 x−2 + c, or
h) 31 e3x + c,
l) ex + c m) −e−x + c, n) sin x + c,
sin 2x + c, c)
x1/2 1/2
d)
i) 71 e7x + c,
o) − cos x + c,
+ c, f) −e−t + c.
c Pearson Education Ltd 2000
