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Table of Integrals - Forms Involving $cosh ax$

The integrals below involve $cosh ax$ .

1) $int cosh ax dx = (sinh ax)/a$

2) $int x*cosh ax dx = (x*sinh ax)/a-(cosh ax)/a^2$

3) $int x^2 cosh ax dx = -(2x cosh ax)/a^2+(x^2/a+2/a^3)sinh ax$

4) $int (cosh ax)/x dx = ln x+(ax)^2/(2*2!)+(ax)^4/(4*4!)+(ax)^6/(6*6!)+...$

5) $int (cosh ax)/x^2 dx = -(cosh ax)/x+a int (sinh ax)/x dx$

**[See integral #4 int the table involving $sinh ax$ ]

6) $int 1/(cosh ax) dx = 2/a tan^-1 e^(ax)$

7) $int x/(cosh ax) dx = 1/a^2{(ax)^2/2-(ax)^4/8+(5(ax)^6)/144+...+((-1)^nE_n(ax)^(2n+2))/((2n+2)(2n)!)+...}$

8) $int cosh^2 ax dx = x/2+(sinh ax*cosh ax)/(2a)$

9) $int x*cosh^2 ax dx = x^2/4+(x sinh 2ax)/(4a)-(cosh 2ax)/(8a^2)$

10) $int 1/(cosh^2 ax) dx = (tanh ax)/a$

11) $int cosh ax*cosh px dx = (sinh(a-p)x)/(2(a-p))+(sinh(a+p)x)/(2(a+p))$

12) $int cosh ax*sin px dx = (a sinh ax*sin px-p cosh ax*cos px)/(a^2+p^2)$

13) $int cosh ax*cos px dx = (a sinh ax*cos px+p cosh ax*sin px)/(a^2+p^2)$

14) $int 1/(cosh ax+1) dx = 1/a tanh((ax)/2)$

15) $int 1/(cosh ax-1) dx = -1/a coth((ax)/2)$

16) $int x/(cosh ax+1) dx = x/a tanh ((ax)/2)-2/a^2 ln cosh ((ax)/2)$

17) $int x/(cosh ax-1) dx = -x/a coth ((ax)/2)+2/a^2 ln sinh ((ax)/2)$

18) $int 1/(cosh ax+1)^2 dx = 1/(2a)tanh((ax)/2)-1/(6a)tanh^3((ax)/2)$

19) $int 1/(cosh ax-1)^2 dx = 1/(2a) coth((ax)/2)-1/(6a) coth^3((ax)/2)$

20) $int 1/(p+q cosh ax) dx = 2/(asqrt(q^2-p^2))tan^-1((qe^(ax)+p)/sqrt(q^2-p^2))$

OR $=1/(asqrt(p^2-q^2))ln((qe^(ax)+p-sqrt(p^2-q^2))/(qe^(ax)+p+sqrt(p^2-q^2)))$

21) $int 1/(p+q cosh ax)^2 dx = (q sinh ax)/(a(q^2-p^2)(p+q cosh ax))-p/(q^2-p^2) int 1/(p+q cosh ax) dx$

22) $int 1/(p^2-q^2 cosh^2 ax) dx = 1/(2apsqrt(p^2-q^2)) ln ((ptanh ax+sqrt(p^2-q^2))/(p tanh ax-sqrt(p^2-q^2)))$

OR $=(-1)/(apsqrt(q^2-p^2))tan^-1((p tanh ax)/sqrt(q^2-p^2))$

23) $int1/(p^2+q^2 cosh^2 ax) dx = 1/(2apsqrt(p^2+q^2)) ln((p tanh ax+sqrt(p^2+q^2))/(p tanh ax-sqrt(p^2+q^2)))$

OR $=1/(apsqrt(p^2+q^2)) tan^-1((p tanh ax)/sqrt(p^2+q^2))$

24) $int x^m cosh ax dx = (x^m sinh ax)/a-m/a int x^(m-1) sinh ax dx$

**[See integral #18 in the table involving $sinh ax$ ]

25) $int cosh^n ax dx = (cosh^(n-1) ax*sinh ax)/(an)+(n-1)/n int cosh^(n-2) ax dx$

26) $int (cosh ax)/x^n dx = (-cosh ax)/((n-1)x^(n-1))+a/(n-1) int (sinh ax)/x^(n-1) dx$

**[See integral #20 in the table involving $sinh ax$ ]

27) $int 1/(cosh^n ax) dx = (sinh ax)/(a(n-1)cosh^(n-1) ax)+(n-2)/(n-1) int 1/(cosh^(n-2) ax) dx$

28) $int x/(cosh^n ax) dx = (x sinh ax)/(a(n-1)cosh^(n-1) ax)+1/((n-1)(n-2)a^2 cosh^(n-2) ax)+(n-2)/(n-1) int x/(cosh^(n-2) ax) dx$

Table of Integrals - Forms Involving coshaxcosh ax