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

Table of Integrals - Forms Involving cosaxcos ax