Lecture Notes in Differential Equations - Bruce E. Shapiro

396 APPENDIX A. TABLE OF INTEGRALS
∫
∫
∫
∫
∫
∫
1
1 + x 2 dx = tan−1 x (A.8)
1
a 2 + x 2 dx = 1 a tan−1 x a
x
a 2 + x 2 dx = 1 2 ln |a2 + x 2 | + c
x 2
a 2 + x 2 dx = x − a tan−1 x a
x 3
a 2 + x 2 dx = 1 2 x2 − 1 2 a2 ln |a 2 + x 2 |
1
ax 2 + bx + c dx = 2
2ax + b
√
4ac − b
2 tan−1 √
4ac − b
2
∫
1
(x + a)(x + b) dx = 1
b − a ln a + x
b + x , a ≠ b
∫
x
(x + a) 2 dx = a + ln |a + x|
a + x (A.15)
(A.9)
(A.10)
(A.11)
(A.12)
(A.13)
(A.14)
∫
x
ax 2 + bx + c dx = 1
2a ln |ax2 + bx + c|
−
b
a √ 2ax + b
4ac − b 2 tan−1 √
4ac − b
2
Integrals with Roots
(A.16)
∫ √x
− adx =
2
3 (x − a)3/2 (A.17)
∫
1
√ x ± a
dx = 2 √ x ± a
(A.18)
∫
1
√ a − x
dx = −2 √ a − x
(A.19)