Table of Integrals - Forms Involving cos ax

The integrals below involve cos ax.

1) int  cos ax  dx = (sin ax)/a

2) int  x cos ax  dx = (cos ax)/a^2+(x sin ax)/a

3) int  x^2 cos ax  dx = (2x)/a^2cos ax+(x^2/a-2/a^3)sin ax

4) int  x^3 cos ax  dx = ((3x^2)/a^2-6/a^4)cos ax+(x^3/a-(6x)/a^3)sin ax

5) int  (cos ax)/x  dx = ln x-(ax)^2/(2*2!)+(ax)^4/(4*4!)-(ax)^6/(6*6!)+...

6) int  (cos ax)/x^2  dx = -(cos ax)/x-aint  (sin ax)/x  dx

**[See integral #5 in the previous table; forms involving sin ax]

7) int  1/(cos ax)  dx = 1/a ln (secax+tan ax)=1/aln tan(pi/4+(ax)/2)

8) int  x/(cos ax)  dx = 1/a^2{(ax)^2/2+(ax)^4/8+(5(ax)^6)/144+...+(E_n(ax)^(2n+2))/((2n+2)(2n )!)+...}

9) int  cos^2 ax  dx = x/2+(sin 2ax)/(4a)

10) int  x*cos^2ax  dx = x^2/4+(x*sin 2ax)/(4a)+(cos 2ax)/(8a^2)

11) int  cos^3 ax  dx = (sin ax)/a-(sin^3ax)/(3a)

12) int  cos^4 ax  dx = (3x)/8+(sin 2ax)/(4a)+(sin 4ax)/(32 a)

13) int  1/(cos^2 ax)  dx = (tan ax)/a

14) int  1/(cos^3 ax)  dx = (sin ax)/(2a*cos^2 ax)+1/(2a) ln tan(pi/4+(ax)/2)

15) int  cos ax*cos px  dx = (sin(a-p)x)/(2(a-p))+(sin(a+p)x)/(2(a+p)

**[If a=+-p, see integral #9 in this table]

16) int  1/(1-cos ax)  dx = -1/acot((ax)/2)

17) int  x/(1-cos ax)  dx = -x/acot((ax)/2)+2/a^2 ln sin((ax)/2)

18) int  1/(1+cos ax)  dx = 1/a tan((ax)/2)

19) int  x/(1+cos ax)  dx = x/a tan((ax)/2)+2/a^2 ln cos((ax)/2)

20) int  1/(1-cos ax)^2  dx = -1/(2a)cot((ax)/2)-1/(6a)cot^3((ax)/2)

21) int  1/(1+cos ax)^2  dx = 1/(2a)tan((ax)/2)+1/(6a)tan^3((ax)/2)

22) int  1/(p+q cos ax)  dx = 2/(asqrt(p^2-q^2)) tan^-1sqrt((p-q)/(p+q))tan(1/2ax)

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

**[If p=+-q, see integrals #16 and #18 in this table]

23) int  1/(p+q cos ax)^2  dx = (q sin ax)/(a(q^2-p^2)(p+q cos ax))-p/(q^2-p^2)int  1/(p+q cos ax)  dx

**[If p=+-q, see integrals #20 and #21 in this table]

24) int  1/(p^2+q^2 cos^2 ax)  dx = 1/(apsqrt(p^2+q^2))tan^-1((p tan ax)/sqrt(p^2+q^2))

25) int  1/(p^2-q^2 cos^2 ax)  dx = 1/(apsqrt(p^2-q^2))tan^-1((p tan ax)/sqrt(p^2-q^2))

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

26) int  x^m cos ax  dx = (x^m sin ax)/a+(mx^(m-1))/a^2 cos ax-(m(m-1))/a^2int  x^(m-2) cos ax  dx

27) int  (cos ax)/x^n  dx = -(cos ax)/((n-1)x^(n-1))-a/(n-1)int  (sin ax)/x^(n-1)  dx

**[See integral #27 in the previous table; forms involving sin ax]

28) int  cos^n ax  dx = (sin ax*cos^(n-1)  ax)/(an)+(n-1)/nint  cos^(n-2) ax  dx

29) int  1/(cos^n ax)  dx = (sin ax)/(a(n-1)cos^(n-1) ax)+(n-2)/(n-1)int  1/(cos^(n-2) ax)  dx

30) int  x/(cos^n ax)  dx = (x sin ax)/(a(n-1)cos^(n-1) ax)-1/(a^2(n-1)(n-2)cos^(n-2) ax)+(n-2)/(n-1)int  x/(cos^(n-2) ax)  dx