MA1S11 Tutorial Sheet 101
15-18 December 2015
Useful facts:
• The indefinite integral: For a function f (x) the indefinite integral is the family of
all anti-derivatives
Z
f (x)dx = F (x) + C
(1)
where C is the arbitrary constant of integration and F 0 (x) = f (x).
• Integration table
R
f (x)
f (x)dx
xn+1
n
x (n 6= −1) n+1 + C
− cos x + C
sin x
cos x
sin x + C
2
tan x + C
sec x
• The definite integral: The area under f (x) from a to b is
Z b
f (x)dx = F (x)]ba = F (b) − F (a)
(2)
a
where F 0 (x) = f (x).
• The Fundamental Theorem of Calculus: The first theorem repeats what is above
Z b
F 0 (x)dx = F (b) − F (a)
(3)
a
and the second theorem
• u-substitution:
d
dx
Z
Z
x
f (t)dt = f (x)
0
f (g(x))g (x)dx =
Z
f (u)du
where u = g(x); another way of saying this is that inside the integral
du
dx = du
dx
For the definite integral replace the x limits with u limits
Z b
Z g(b)
0
f (g(x))g (x)dx =
f (u)du
a
1
(4)
a
g(a)
Stefan Sint, sint@maths.tcd.ie, see also http://www.maths.tcd.ie/~sint/MA1S11.html
1
(5)
(6)
(7)
Questions
1. (4) Determine the following indefinite integrals
√
Z
(1 + x)20
√
a)
dx
x
Z
b)
tan(2x) sec2 (2x)dx
Z
√
c)
x2 3 + xdx
2. (4) Calculate the definite integrals
Z π/2
cos(x)dx
a)
0
Z 2
b)
y 2 − y −2 dy
and
1
Z 5
(1 + w)(2w + w2 )5 dw
c)
Z
1
y 2 − y −2 dy
2
−1
Extra Questions
The questions are extra; you don’t need to do them in the tutorial class.
3. Calculate using u-substitution
Z
π/2
cos(x) cos (sin x)dx
−π
4. Calculate the definite integrals
Z
π/2
a)
sin(x) cos(x)dx
0
Z
π/4
b)
sin(x) cos(x)dx
0
Z
π/2
c)
sin(x) cos(x)dx
π/4
and verify the property of the definite integral
Z b
Z c
Z b
f (x)dx =
f (x)dx +
f (x)dx
a
a
with f (x) = sin(x) cos(x), a = π/4, b = 0, c = π/2.
2
c