Math formulas for definite integrals of trigonometric functions

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Math Formulas: Definite integrals of trig
functions
Note: In the following formulas all letters are positive.
Basic formulas
π/2
Z
Z
2
π/2
sin x dx =
1.
cos2 x dx =
0
0

p>0
 π/2
sin(px)
0
p=0
dx =

x
−π/2 p < 0
Z ∞
πp
sin2 px
=
2
x
2
0
Z ∞
1 − cos(px)
πp
dx =
x2
2
0
∞
Z
2.
0
3.
4.
Z
π
4
∞
5.
0
∞
cos(px) − cos(qx)
q
dx = ln
x
p
π(q − p)
cos(px) − cos(qx)
dx =
2
x
2
0
Z 2π
dx
2π
=√
2
a + b sin x
a − b2
0
Z 2π
dx
2π
=√
2
a + b cos(x)
a − b2
0
r
Z ∞
Z ∞
1 π
2
2
sin ax dx =
cos(ax ) dx =
2 2a
0
0
r
Z ∞
Z ∞
sin x
cos x
π
√ dx =
√ dx =
2
x
x
0
0
Z ∞
sin3 x
3π
dx =
3
x
8
0
Z ∞
sin4 x
π
dx =
4
x
3
0
Z ∞
π
tan x
dx =
x
2
0
Z π/2
dx
arccos(b/a)
= √
a + b cos x
a2 − b2
0
Z
6.
7.
8.
9.
10.
11.
12.
13.
14.
Advanced formulas
Z
π
0
Z π
cos(mx) · cos(nx) dx =
16.
0
Z
0
π/2
m, n integers and m 6= n
m, n integers and m = n
0
π/2
m, n integers and m 6= n
m, n integers and m = n
π
0
m, n integers and m + n odd
2m/(m2 − n2 )
m, n integers and m + n even
Z π/2
Z π/2
1 · 3 · 5 . . . 2m − 1 π
sin2m x dx =
cos2m x dx =
2 · 4 · 6 . . . 2m 2
0
0
sin(mx) · cos(nx) dx =
17.
0
18.
sin(mx) · sin(nx) dx =
15.
1
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Z
19.
π/2
2m+1
sin
Z
x dx =
0
π/2
cos2m+1 x dx =
0
Z
20.
π
sin2p−1 x cos2q−1 x dx =
0
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
Γ(p) Γq
2 Γ(p + q)

p>q>0
 0
sin(px) · cos(qx)
π/2 0 < p < q
dx =

x
0
π/4 p = q > 0
Z ∞
sin(px) · sin(qx)
π p/2 0 < p ≤ q
dx =
π q/2 p ≥ q > 0
x2
0
Z ∞
cos(mx)
π −ma
dx =
e
2 + a2
x
2a
0
Z ∞
π
x sin(mx)
dx = e−ma
2
2
x +a
2
0
Z ∞
π
sin(mx)
dx = 2 1 − e−ma
2 + a2 )
x
(x
2a
0
Z 2π
Z 2π
dx
dx
2π a
=
= 2
2
2
(a
+
b
sin
x)
(a
+
b
cos
x)
(a
−
b2 )3/2
0
0
Z 2π
dx
2π
=
, 0<a<1
2
1
−
2a
cos
x
+
a
1
−
a2
0
π
Z π
x sin x dx
|a| < 1
a ln(1 + a)
=
2
π ln(1 + a1 ) |a| > 1
0 1 − 2a cos x + a
Z π
cos(mx) dx
πam
=
, a2 < 1
2
1 − a2
0 1 − 2a cos x + a
Z ∞
π
1
Γ(1/n) sin
, n>1
sin(axn ) dx =
1/n
2n
na
0
Z ∞
1
π
cos(axn ) dx =
Γ(1/n) cos
, n>1
1/n
2n
na
0
Z ∞
sin x
π
dx =
, 0<p<1
p
x
2 Γ(p) sin(pπ/2)
0
Z ∞
cos x
π
dx =
, 0<p<1
p
x
2
Γ(p)
cos(pπ/2)
0
r Z ∞
b2
b2
1 π
cos
− sin
sin(ax2 ) cos(2bx) dx =
2 2a
a
a
0
r Z ∞
1 π
b2
b2
cos(ax2 ) cos(2bx) dx =
+ sin
cos
2 2a
a
a
0
Z ∞
dx
π
m dx =
1 + tan x
4
0
Z
21.
2 · 4 · 6 . . . 2m
1 · 3 · 5 . . . 2m + 1
∞
2