MATH1010 Further Exercise on Integration

MATH1010 Further Exercise on Integration
1. Evaluate the definite integrals below:
(a)
Z
4
|x(2 − x)|dx
0
Z
(b)
2
|x2 − 3x + 2|dx
0
Z
(c)
2
|x3 − x2 − 2x|dx
−1
2. Evaluate the definite/indefinite integrals below:
(a)
Z
(e)
Z
x(x + 2)99 dx
(f)
(i)
Z
xdx
√
x+9
Z
(l)
Z
x(x2 + 2)99 dx
2
(j)
3
Z
(b)
Z
4
3
5
√
1
x
dx
25 − x2
xdx
4x + 5
1
3
x (1 + 3x ) dx
Z
(k)
0
2
Z
(m)
3
0
(x2
Z
x
√
dx
3x2 + 1
Z
(c)
(g)
1
2
2
3x + 9x − 12x + 4
dx
9x5 − 12x4 + 4x3
1
√
√
x x − 1dx
1
−1
(d)
(h)
Z
Z
2
0
x2
√
dx
9 − x3
√
(x + 2) x − 1dx
1 + x2
dx
1 + 9x2
2(x − 1)
dx
+ 3)(x + 1)2
3. Evaluate the definite/indefinite integrals below:
(a)
(d)
3π
4
Z
π
4
Z
(g)
Z
(j)
Z
π
2
(b)
(sin(x) + cos(x))2 dx
(e)
π
sin5 (x)dx
Z
(h)
0
sin3 (x) cos4 (x)dx
π
2
(l)
(o)
Z
2
x sin (x )dx
Z
π
| sin(2x) − sin(x)|dx
0
(1 + cos(x))4 dx
(i)
Z
(f)
Z
π
2
π
2
cos(3x) cos(x)dx
sin6 (x) cos2 (x)dx
0
cos6 (x)dx
0
csc4 (x) cos(x)dx
2dx
(n)
cot(x/2) + tan(x/2)
Z π
||sin(2x)| − sin(x)| dx
(p)
Z
Z
(c)
0
0
(k)
(m)
cos3 (x) sin2 (x)dx
cos5 (x) sin2 (x)dx
0
2
π
2
Z
0
π
2
Z
Z
(sin(x) − cos(x))dx
Z
√
2π
x3 cos2 (x2 )dx
0
0
4. Evaluate the definite/indefinite integrals below:
5
(a)
Z
(d)
Z
(g)
(i)
(l)
(o)
√
1 + 3xdx
Z
(b)
0
2(x2 + x)e2x dx
3
0
(e)
3x + 4
√
dx
(c)
x+1
Z 1
1
ln(x)dx
+
x x2
Z
√
x
dx
1+x
Z
(f)
e−5x sin(4x)dx
x5 dx
. (Try not to ‘break up’ x3 − 1.)
x3 − 1
Z
Z 1 3
Z
x +x
1 − 2x
x+1
dx
(j)
dx
(k)
dx
2
2
3
2
3
x (x + 1)
x +x +x+1
0 x +1
Z
Z π2
Z π4
24dx
dx
4
sec (x)dx
(n)
(m)
2
x3 − x2 − 9x + 9
(1
+
cos(x))
0
0
Z
Z π4
Z π2
x sin2 (x)dx
(p)
(1 + sec(x)) tan(x)dx
(q)
tan3 (x)dx
dx
2
x − 3x + 2
Z
(h)
Z
0
(r)
(u)
Z
0
π
3
(1 + tan6 (x))dx
(s)
0
Z
dx
(1 + cos(x)) sin(x)
(v)
Z
π
6
sin(x) tan(x)dx
(t)
Z
cos2 (x) cot(x)dx
(w)
Z
cos2 (2x)
dx
sin (x) cos2 (x)
0
Z
π
6
0
tan(x)dx
1 + sin2 (x)
1
4
(x)
5. (a)
1 + cos(x)
dx
x + sin(x)
Z
x + sin(x)
dx
1 + cos(x)
Z
(y)
i. Express 5 + 4 cos(x) + sin(x) in the form a cos2
ii. Hence, or otherwise, compute
Z
π
2
0
(z)
x
2
Z
π
2
0
+ b sin2
dx
.
5 + 4 cos(x) + 3 sin(x)
s
1 − sin(x)
dx
1 + sin(x)
x
2
, where a, b are constants.
(b) Apply the above method, or otherwise, to evaluate the definite/indefinite integrals below:
Z
dx
i.
1 + sin(x) + cos(x)
Z
dx
ii.
4 cos(x) + 3 sin(x)
Z π2
dx
iii.
2
+
cos(x)
0
π
2
dx
.
3 + 2 sin(x) + cos(x)
Z π2
(2 sin(x) + cos(x))dx
(b) Hence, or otherwise, evaluate
.
3 + 2 sin(x) + cos(x)
0
6. (a) Evaluate
7. Show that
Z
π
0
Z
0
x sin(x)
dx =
1 + cos2 (x)
Z
π
0
(π − x) sin(x)
dx. Hence, or otherwise, evaluate both definite integrals.
1 + cos2 (x)
Z π2
sin4 (x)
cos4 (x)
dx
=
dx.
4
4
4
π sin (x) + cos (x)
sin (x) + cos4 (x)
0
2
Z π
sin4 (x)
(b) Hence, or otherwise, compute
dx.
4
4
0 sin (x) + cos (x)
Z π
Z π
x sin4 (x)
(π − x) sin4 (x)
(c) Show that
dx
=
dx. Hence, or otherwise, evaluate both definite
4
4
4
4
0 sin (x) + cos (x)
0 sin (x) + cos (x)
integrals.
8. (a) Show that
Z
π
9. Let n be a positive integer.
(a) Show that cos(x) + cos(3x) + cos(5x) + · · · + cos((2n − 1)x) =
(b) Evaluate
(c) Evaluate
Z
Z
π
3
sin(2nx)
whenever sin(x) 6= 0.
2 sin(x)
sin(2nx)
dx
sin(x)
π
6
π
3
π
6
(sin(x) + 3 sin(3x) + 5 sin(5x) + · · · + (2n − 1) sin((2n − 1)x)) dx.
10. For any positive integer n, define In =
(a) Show that whenever n ≥ 2, In =
Z
1
0
√
xn 1 − xdx.
2n
In−1 .
2n − 3
(b) Evaluate I10 .
11. For any positive integers m, n, define Im,n =
Z
0
Im,n =
π
2
sinm (x) cosn (x)dx. Show that, whenever m, n ≥ 2,
n−1
n−1
m−1
m−1
Im−2,n =
Im,n−2 =
Im+2,n−2 =
Im−2,n+2 .
m+n
m+n
m+1
n+1
2
dn n −x
(x e ) on R is a polynomial function
dxn
of degree n with leading coefficient 1. (Hint: Apply Leibniz’s Rule.)
12. (a) Let n be a non-negative integer. Show that the function (−1)n ex
(b) For each non-negative integer, define the function Ln : R −→ R by Ln (x) = (−1)n ex
dn n −x
(x e ).
dxn
Suppose m, n are positive integers.
i. Show that
lim
x→+∞
Z
x
−t
Ln (t)Lm (t)e dt = (−1)
n−1
lim
x→+∞
0
Z
x
0
ii. Hence, or otherwise, show that
lim
x→+∞
Z
x
0

 (n!)2
Ln (t)Lm (t)e−t dt =
 0
dn−1 n −t ′
t e
Lm (t)dt,
dtn−1
if
n=m
if
n 6= m
13. Let f : R −→ R be a function. Suppose f is differentiable on R and f (0) = f (1) = 0.
Z 1
Z 1
1
xf (x)f ′ (x)dx = − .
(f (x))2 dx = 1. Show that
Further suppose
2
0
0
14. Evaluate the limits below.
(a)
(d)
1
lim
h→0 h
lim
h→0+
Z
sin(h)
sin(
0
1
ln(1 + h)
Z
p
t2
3h+2
2
+
t4 )dt
p
15. Define f : (0, +∞) −→ R by f (x) =
(a) Show that f (x) =
1
x
Z
x2
1
1
lim
h→0 h sin(h)
(b)
Z
h2
t2
e dt
(c)
0
1
lim
h→0 h
Z
(d)
Z
h
−h
t6 + 2t4 + 3t2 + 4dt
Z
x
x−1
q
3
sin5 (t) dt
√
cos( xt)dt for any x ∈ (0, +∞).
√
cos( u)du for any x ∈ (0, +∞).
′
(b) Find the value of f (1).
16. Evaluate the first derivative of the functions of x below.
(a)
(e)
Z
Z
x3
x
−2
p
sin(x)
0
t4 + t + 1dt.
cos(x2 ) cos(t2 )
dt
2+t
(b)
Z
(f)
2x
2
e3t dt
(c)
x
Z
x2
sin
x
t
x2
dt
Z
x
7
−x
| cos(t)| 2 dt
(g)
Z
x
20
Z
u
10
3
0
dt
4
1 + t + sin4 (t)
17. Let f : [0, 1] −→ R be a function. Suppose f is continuous on [0, 1]. Further suppose that
for any x ∈ [0, 1]. Show that f (x) = 0 for any x ∈ [0, 1].
sin(x)
Z
cos2 (t2 )
dt
2 + t2
du
x
f (t)dt =
0
Z
1
f (t)dt
x

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