Lecture Notes in Differential Equations - Bruce E. Shapiro

404 APPENDIX A. TABLE OF INTEGRALS
Products of Trigonometric Functions and Monomials
∫
x cos xdx = cos x + x sin x
(A.93)
∫
x cos axdx = 1 a 2 cos ax + x sin ax
a (A.94)
∫
x 2 cos xdx = 2x cos x + ( x 2 − 2 ) sin x
(A.95)
∫
x 2 cos axdx =
2x cos ax
a 2 + a2 x 2 − 2
a 3 sin ax (A.96)
∫
x n cosxdx = − 1 2 (i)n+1 [Γ(n + 1, −ix) + (−1) n Γ(n + 1, ix)]
(A.97)
∫
x n cosaxdx = 1 2 (ia)1−n [(−1) n Γ(n + 1, −iax) − Γ(n + 1, ixa)]
(A.98)
∫
x sin xdx = −x cos x + sin x
(A.99)
∫
∫
x cos ax sin ax
x sin axdx = − +
a a 2
x 2 sin xdx = ( 2 − x 2) cos x + 2x sin x
(A.100)
(A.101)
∫
x 2 sin axdx = 2 − a2 x 2
cos ax +
a 3
2x sin ax
a 2
(A.102)
∫
x n sin xdx = − 1 2 (i)n [Γ(n + 1, −ix) − (−1) n Γ(n + 1, −ix)]
(A.103)