Math Formulas: De nite integrals of trig functions


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Math Formulas: Definite integrals of trig functions

Note: In the following formulas all letters are positive.
Basic formulas

1. π/2 sin2 x dx = π/2 cos2 x dx = π

0

0

4



∞ sin(px)

 π/2 p > 0

2.

dx = 0

p=0

0

x

 −π/2 p < 0

∞ sin2 px π p

3.

=

0

x2

2

∞ 1 − cos(px)

πp

4.

dx =

0

x2

2

∞ cos(px) − cos(qx)

q

5.

dx = ln

0

x

p

∞ cos(px) − cos(qx)

π(q − p)

6.

dx =

0

x2

2

2π dx



7.

=√

0 a + b sin x

a2 − b2



dx



8.

=√

0 a + b cos(x)

a2 − b2

9. ∞ sin ax2 dx = ∞ cos(ax2) dx = 1 π

0

0

2 2a

∞ sin x

∞ cos x

π

10.

√ dx =

√ dx =

0

x

0

x

2

∞ sin3 x



11.

dx =

0 x3

8

∞ sin4 x

π

12.

dx =

0 x4

3

∞ tan x

π

13.

dx =

0x

2

π/2

dx

arccos(b/a)

14.

=√

0 a + b cos x

a2 − b2

Advanced formulas

π

0

m, n integers and m = n

15. 0 sin(mx) · sin(nx) dx = π/2 m, n integers and m = n

π

0

m, n integers and m = n

16. 0 cos(mx) · cos(nx) dx = π/2 m, n integers and m = n

π

0

m, n integers and m + n odd

17.

sin(mx) · cos(nx) dx = 2m/(m2 − n2) m, n integers and m + n even

0

18. π/2 sin2m x dx = π/2 cos2m x dx = 1 · 3 · 5 . . . 2m − 1 π

0

0

2 · 4 · 6 . . . 2m 2

1

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19. π/2 sin2m+1 x dx = π/2 cos2m+1 x dx = 2 · 4 · 6 . . . 2m

0

0

1 · 3 · 5 . . . 2m + 1

20. π sin2p−1 x cos2q−1 x dx = Γ(p) Γq

0

2 Γ(p + q)



∞ sin(px) · cos(qx)

 0 p>q>0

21.

dx = π/2 0 < p < q

0

x

 π/4 p = q > 0

∞ sin(px) · sin(qx)

π p/2 0 < p ≤ q

22. 0 x2 dx = π q/2 p ≥ q > 0

23. ∞ cos(mx) dx = π e−ma

0 x2 + a2

2a

24. ∞ x sin(mx) dx = π e−ma

0 x2 + a2

2

25. ∞ sin(mx) dx = π 1 − e−ma

0 x (x2 + a2)

2a2



dx



dx

2π a

26.

=

=

0 (a + b sin x)2 0 (a + b cos x)2 (a2 − b2)3/2



dx



27.

=

, 0
0 1 − 2a cos x + a2 1 − a2

π x sin x dx

π ln(1 + a) |a| < 1

28. 0 1 − 2a cos x + a2 = πa ln(1 + a1 ) |a| > 1

29. π cos(mx) dx = πam , a2 < 1 0 1 − 2a cos x + a2 1 − a2

30. ∞ sin(axn) dx = 1 Γ(1/n) sin π , n > 1

0

na1/n

2n

31. ∞ cos(axn) dx = 1 Γ(1/n) cos π , n > 1

0

na1/n

2n

∞ sin x

π

32.

dx =

, 0
0 xp

2 Γ(p) sin(pπ/2)

∞ cos x

π

33.

dx =

, 0
0 xp

2 Γ(p) cos(pπ/2)

34. ∞ sin(ax2) cos(2bx) dx = 1 π cos b2 − sin b2

0

2 2a

a

a

35. ∞ cos(ax2) cos(2bx) dx = 1 π cos b2 + sin b2

0

2 2a

a

a

∞ dx

π

36. 0 1 + tanm x dx = 4

2

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Math Formulas: De nite integrals of trig functions