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`