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CEGEP CHAMPLAIN - ST. LAWRENCE
201-203-RE: Integral Calculus
Patrice Camiré
Problem Sheet #9
Trigonometric Integrals
1. Evaluate the following trigonometric integrals: sin(x) and/or cos(x) with at least one odd power.
2
Z
2
sec(x) dx = ln | sec(x) + tan(x)| + C
sin (x) + cos (x) = 1
d
sin(x) = cos(x)
dx
Z
Z
sin(x) cos(x) dx
(a)
Z
(b)
Z
(c)
Z
(d)
Z
(e)
Z
(f)
Z
(g)
d
cos(x) = − sin(x)
dx
(h)
Z
sin(x) cos2 (x) dx
(i)
Z
sin3 (x) cos2 (x) dx
(j)
Z
cos5 (x) sin2 (x) dx
(k)
sin8 (x) cos3 (x) dx
sin3 (x) dx
Z
cos3 (x) dx
Z
sin4 (x)
dx
cos(x)
Z
sin3 (4x) dx
Z
sin3 (2x) cos3 (2x) dx
Z
cos5 (x) sin3 (x) dx
(o)
sin(x)
dx
cos3 (x)
(p)
cos3 (x)
dx
sin(x)
(q)
sin3 (x)
p
dx
cos(x)
Z
(r)
cos5 (x) sin4 (x) dx
(s)
Z
sin3 (x)
dx
cos(x)
Z p
3
(t)
cos(x) sin3 (x) dx
Z
sin2 (x)
dx
cos(x)
(l)
(m)
cos5 (x) dx
cos(x)
dx
1 + sin(x)
(n)
Z
(u)
sin5
x
2
dx
2. Evaluate the following trigonometric integrals: sin(x) and/or cos(x) with only even powers.
cos2 (x) =
Z
(a)
Z
(b)
sin2 (x) dx
2
cos (x) dx
1 + cos(2x)
2
Z
(c)
Z
(d)
sin2 (x) =
sin2 (x) cos2 (x) dx
2
sin (3x) dx
1 − cos(2x)
2
Z
cos2 (5x) dx
Z
sin2 (3x + 1) dx
(e)
(f)
3. Evaluate the following trigonometric integrals: tan(x) and/or sec(x).
2
2
tan (x) + 1 = sec (x)
d
tan(x) = sec2 (x)
dx
Z
(a)
sec (x) dx
Z
(b)
2
tan (x) dx
Z
(c)
Z
(d)
sec4 (x) dx
4
tan (x) dx
Z
(e)
Z
(f)
Z
(g)
Z
(h)
2
tan(2x) sec2 (2x) dx
sec(x) tan3 (x) dx
tan3 (x) sec2 (x) dx
sec(x) tan(x)
dx
sec(x) − 1
Z
Z
tan(x) dx = ln | sec(x)| + C
sec(x) dx = ln | sec(x) + tan(x)| + C
d
sec(x) = sec(x) tan(x)
dx
Z
(i)
5
tan (x) dx
Z
(j)
Z
(k)
(m)
Z
(n)
Z
(o)
sec6 (x) dx
Z
tan3 (x)
dx
sec(x)
Z
tan2 (x) sec4 (x) dx
Z
tan3 (x) sec5 (x) dx
Z
tan5 (x) sec(x) dx
Z
tan3 (x)
dx
sec2 (x)
Z
sec(x)
dx
tan2 (x)
Z
sin2 (x)
dx
cos6 (x)
Z
sin4 (x)
dx
cos2 (x)
tan3 (x) dx
3
tan5 (x) sec7 (x) dx
(s)
(t)
tan2 (x)
dx
sec(x)
8
(u)
sec (x) tan(x) dx
Z
(p)
Z
(r)
tan(x) sec (x) dx
Z
sec4 (x) dx
(q)
tan8 (x) sec4 (x) dx
Z
(l)
Z
3
tan (x) sec(x) dx
4. Evaluate the following trigonometric integrals: miscellaneous.
Z
Z
1 − sin(x)
tan(x)
dx
(e)
dx
(a)
sec(x)
cos(x)
Z
Z
tan4 (x)
sin2 (x)
(f)
dx
(b)
dx
cos4 (x)
cos4 (x)
Z
Z
1
4
(g)
dx
(c)
tan(x) cos (x) dx
2
sin (x) sec3 (x)
√
Z
Z
tan3 (x)
sin3 ( x)
√
(d)
dx
(h)
dx
cos3 (x)
x
(v)
(i)
(j)
(k)
(l)
Answers
1. (a)
(b)
(c)
(d)
(e)
(f)
(g)
(h)
(i)
(j)
(k)
1
sin2 (x) + C
2
1
− cos3 (x) + C
3
1
1
cos5 (x) − cos3 (x) + C
5
3
2
1
1
3
sin (x) − sin5 (x) + sin7 (x) + C
3
5
7
1
1
sin9 (x) −
sin11 (x) + C
9
11
1
cos3 (x) − cos(x) + C
3
2
1
sin(x) − sin3 (x) + sin5 (x) + C
3
5
ln |1 + sin(x)| + C
1
sec2 (x) + C
2
1
ln | sin(x)| − sin2 (x) + C
2
2
cos5/2 (x) − 2 cos1/2 (x) + C
5
x 1
− sin(2x) + C
2 4
x 1
(b)
+ sin(2x) + C
2 4
2. (a)
1
2
1
sin5 (x) − sin7 (x) + sin9 (x) + C
5
7
9
cos2 (x)
(m)
− ln | cos(x)| + C
2
(n) ln | sec(x) + tan(x)| − sin(x) + C
(l)
(o) sin(x) −
(p) ln | sec(x) + tan(x)| − sin(x) −
(q)
(r)
(s)
(t)
(u)
sin3 (x)
+C
3
cos3 (4x) cos(4x)
−
+C
12
4
sin4 (2x) sin6 (2x)
−
+C
8
12
cos8 (x) cos6 (x)
−
+C
8
6
3
3
cos10/3 (x) − cos4/3 (x) + C
10
4
x 4
x 2
x
−2 cos
+ cos3
− cos5
+C
2
3
2
5
2
1
x
−
sin(4x) + C
8 32
x sin(6x)
(d)
−
+C
2
12
(c)
3. (a) tan(x) + C
(b) tan(x) − x + C
1
(c) tan(x) + tan3 (x) + C
3
1
3
(d) tan (x) − tan(x) + x + C
3
1
(e) tan2 (2x) + C
4
1
(f) sec3 (x) − sec(x) + C
3
1
(g) tan4 (x) + C
4
(h) ln | sec(x) − 1| + C
1
1
(i) tan4 (x) − tan2 (x) + ln | sec(x)| + C
4
2
1
1
(j) tan9 (x) +
tan11 (x) + C
9
11
1
(k) tan2 (x) − ln | sec(x)| + C
2
1
(l) sec3 (x) + C
3
sin3 (x)
+C
3
x sin(10x)
+
+C
2
20
x sin(6x + 2)
(f)
−
+C
2
12
(e)
1
2
1
sec11 (x) − sec9 (x) + sec7 (x) + C
11
9
7
(n) ln | sec(x) + tan(x)| − sin(x) + C
(m)
sec8 (x)
+C
8
sec3 (x)
− sec(x) + C
(p)
3
tan3 (x)
(q) tan(x) +
+C
3
tan5 (x) 2
(r)
+ tan3 (x) + tan(x) + C
5
3
(s) sec(x) + cos(x) + C
(o)
tan3 (x) tan5 (x)
+
+C
3
5
sec7 (x) sec5 (x)
(u)
−
+C
7
5
sec5 (x) 2
(v)
− sec3 (x) + sec(x) + C
5
3
(t)
4. (a) − cos(x) + C
1
(b) tan3 (x) + C
3
1
(c) − cos4 (x) + C
4
1
1
(d) − sec3 (x) + sec5 (x) + C
3
5
(e) ln |1 + sin(x)| + C
1
1
(f) tan5 (x) + tan7 (x) + C
5
7
(g) − csc(x) − sin(x) + C
(h)
√
√
2
cos3 ( x) − 2 cos( x) + C
3
cos2 (x)
− ln | cos(x)| + C
2
1
(j) −
+C
sin(x)
(i)
(k)
tan3 (x) tan5 (x)
+
+C
3
5
3
sin(2x)
(l) tan(x) − x −
+C
2
4