www.mathportal.org 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 www.mathportal.org 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