Processing math: 100%

Table of Integrals - Forms Involving $a^2-x^2, x^2lta^2$

The integrals below involve $a^2-x^2$ where $x^2lta^2$

1) $int 1/(a^2-x^2) dx = 1/(2a) ln ((a+x)/(a-x))$

OR $=1/a tanh^-1 (x/a)$

2) $int x/(a^2-x^2) dx = -1/2 ln (a^2-x^2)$

3) $int x^2/(a^2-x^2) dx = -x+a/2 ln ((a+x)/(a-x))$

4) $int x^3/(a^2-x^2) dx = -x^2/2-a^2/2 ln (a^2-x^2)$

5) $int 1/(x(a^2-x^2)) dx = 1/(2a^2) ln (x^2/(a^2-x^2))$

6) $int 1/(x^2(a^2-x^2)) dx = -1/(a^2x)+1/(2a^3) ln ((a+x)/(a-x))$

7) $int 1/(x^3(a^2-x^2)) dx = -1/(2a^2x^2)+1/(2a^4) ln (x^2/(a^2-x^2))$

8) $int 1/(a^2-x^2)^2 dx = x/(2a^2(a^2-x^2))+1/(4a^3) ln ((a+x)/(a-x))$

9) $int x/(a^2-x^2)^2 dx = 1/(2(a^2-x^2)$

10) $int x^2/(a^2-x^2)^2 dx = x/(2(a^2-x^2))-1/(4a) ln ((a+x)/(a-x))$

11) $int x^3/(a^2-x^2)^2 dx = a^2/(2(a^2-x^2))+1/2 ln (a^2-x^2)$

12) $int 1/(x(a^2-x^2)^2) dx = 1/(2a^2(a^2-x^2))+1/(2a^4) ln (x^2/(a^2-x^2))$

13) $int 1/(x^2(a^2-x^2)^2) dx = (-1)/(a^4x)+x/(2a^4(a^2-x^2))+3/(4a^5) ln ((a+x)/(a-x))$

14) $int 1/(x^3(a^2-x^2)^2) dx = (-1)/(2a^4x^2)+1/(2a^4(a^2-x^2))+1/(a^6) ln (x^2/(a^2-x^2))$

15) $int 1/(a^2-x^2)^n dx = x/(2(n-1)a^2(a^2-x^2)^(n-1))+(2n-3)/((2n-2)a^2) int 1/(a^2-x^2)^(n-1) dx$

16) $int x/(a^2-x^2)^n dx = 1/(2(n-1)(a^2-x^2)^(n-1))$

17) $int 1/(x(a^2-x^2)^n) dx = 1/(2(n-1)a^2(a^2-x^2)^(n-1))+1/(a^2) int 1/(x(a^2-x^2)^(n-1)) dx$

18) $int x^m/(a^2-x^2)^n dx = a^2 int x^(m-2)/(a^2-x^2)^n dx- int x^(m-2)/(a^2-x^2)^(n-1) dx$

19) $int 1/(x^m(a^2-x^2)^n) dx = 1/a^2 int 1/(x^m(a^2-x^2)^(n-1)) dx +1/a^2 int 1/(x^(m-2)(a^2-x^2)^n) dx$

Table of Integrals - Forms Involving a2-x2,x2<a2a^2-x^2, x^2lta^2