# Integration Formulas (Table of the Indefinite Integrals)

## Related Calculator: Integral (Antiderivative) Calculator with Steps

Below is a table of Indefinite Integrals. With this table and integration techniques, you will be able to find majority of integrals.

It is also worth noting that unlike derivative (we can find derivative of any function), we can't find integral of any function: this means that we can't find integral in terms of functions we know.

Examples of such functions are int e^(x^2)dx , int (sin(x))/xdx , int sqrt(x^3+1)dx etc.

 Basic Forms int (af(x)+bg(x))dx=a int f(x)dx+b int g(x)dx where a and b are constants int udv=uv-int vdu (Integration by Parts) int x^n dx={((x^(n+1))/(n+1)+C if n!=-1),(ln|x|+C if n=-1):} int a^x dx=(a^x)/(ln(a))+C int e^x dx=e^x+C int sin(x)dx=-cos(x)+C int cos(x)dx=sin(x)+C int sec^2 (x)dx=tan(x)+C int csc^2(x)dx=-cot(x)+C int sec(x)tan(x)dx=sec(x)+C int csc(x)cot(x)dx=-csc(x)+C int tan(x)dx=-ln|cos(x)|+C=ln|sec(x)|+C int cot(x)dx=ln|sin(x)|+C int sec(x)dx=ln|sec(x)+tan(x)|+C int csc(x)dx=ln|csc(x)-cot(x)|+C int (dx)/sqrt(a^2-x^2)=arcsin(x/a)+C int (dx)/(a^2+x^2)=1/a arctan(x/a)+C int (dx)/(xsqrt(x^2-a^2))=1/a text(arcsec)(x/a)+C int (dx)/(a^2-x^2)=1/(2a)ln|(x+a)/(x-a)|+C int (dx)/(x^2-a^2)=1/(2a)ln|(x-a)/(x+a)|+C Exponential and Logarithmic Forms int xe^(ax)dx=1/(a^2)(ax-1)e^(ax)+C int x^n e^(ax)dx=1/a x^n e^(ax)- n/a int x^(n-1)e^(ax)dx int e^(ax)sin(bx)dx=(e^(ax))/(a^2+b^2)(asin(bx)-bcos(bx))+C int e^(ax)cos(bx)dx=(e^(ax))/(a^2+b^2)(acos(bx)+bsin(bx))+C int ln(x)dx=x(ln(x)-1)+C int x^n ln(x)dx=(x^(n+1))/((n+1)^2)((n+1)ln(x)-1)+C int 1/(xln(x))dx=ln|ln(u)|+C Trigonometric Forms int sin^2(x)dx=1/2 x-1/4 sin(2x)+C int cos^2(x)dx=1/2 x+1/4 sin(2x)+C int tan^2(x)dx=tan(x)-x+C int cot^2(x)dx=-cot(x)-x+C int sin^3(x)dx=-1/3 (2+sin^2(x))cos(x)+C int cos^3(x)dx=1/3 (2+cos^2(x))sin(x)+C int tan^3(x)dx=1/2 tan^2(x)+ln|cos(x)|+C int cot^3(x)dx=-1/2 cot^2(x)-ln|sin(x)|+C int sec^3(x)dx=1/2 sec(x)tan(x)+1/2 ln|sec(x)+tan(x)|+C int csc^3(x)dx=-1/2 csc(x)cot(x)+1/2 ln|csc(x)-cot(x)|+C int sin^n(x)dx=-1/n sin^(n-1)(x)cos(x)+(n-1)/n int sin^(n-2)(x)dx int cos^n(x)dx=1/n cos^(n-1)(x)sin(x)+(n-1)/n int cos^(n-2)(x)dx int tan^n (x)dx=1/(n-1) tan^(n-1)(x)-int tan^(n-2)(x)dx int cot^n (x)dx=-1/(n-1) cot^(n-1)(x)-int cot^(n-2)(x)dx int sec^n (x)dx=1/(n-1) tan(x) sec^(n-2)(x)+(n-2)/(n-1) int sec^(n-2) (x)dx int csc^n (x)dx=-1/(n-1)cot(x)csc^(n-2)(x)+(n-2)/(n-1) int csc^(n-2)(x)dx int sin(ax)sin(bx)dx=(sin((a-b)x))/(2(a-b))-(sin((a+b)x))/(2(a+b))+C int cos(ax)cos(bx)dx=(sin((a-b)x))/(2(a-b))+(sin((a+b)x))/(2(a+b))+C int sin(ax)cos(bx)dx=-(cos((a-b)x))/(2(a-b))-(cos((a+b)x))/(2(a+b))+C int xsin(x)dx=sin(x)-xcos(x)+C int xcos(x)dx=cos(x)+xsin(x)+C int x^n sin(x)dx=-x^n cos(x)+n int x^(n-1)cos(x)dx int x^n cos(x)dx=x^n sin(x)-n int x^(n-1) sin(x)dx int sin^n (x) cos^m (x)dx=-(sin^(n-1)(x)cos^(m+1)(x))/(n+m)+(n-1)/(n+m) int sin^(n-2)(x)cos^m(x)dx= =(sin^(n+1)(x)cos^(m-1)(x))/(n+m)+(m-1)/(n+m) int sin^n(x) cos^(m-2)(x)dx Inverse Trigonometric Forms int arcsin(x)dx=xarcsin(x)+sqrt(1-x^2)+C int arccos(x)dx=xarccos(x)-sqrt(1-x^2)+C int arctan(x)dx=x arctan(x)-1/2 ln(1+x^2)+C int x arcsin(x)dx=(2x^2-1)/4 arcsin(x)+(xsqrt(1-x^2))/4+C int x arccos(x)dx=(2x^2-1)/4 arccos(x)-(xsqrt(1-x^2))/4+C int x arctan(x)dx=(x^2+1)/2 arctan(x)-x/2+C int x^n arcsin(x)dx=1/(n+1)(x^(n+1)arcsin(x)-int (x^(n+1))/(sqrt(1-x^2))dx) , n!=-1 int x^n arccos(x)dx=1/(n+1)(x^(n+1)arccos(x)+int (x^(n+1))/(sqrt(1-x^2))dx) , n!=-1 int x^n arctan(x)dx=1/(n+1)(x^(n+1)arctan(x)-int (x^(n+1))/(1+x^2)dx) , n!=-1 Hyperbolic Forms int sinh(x)dx=cosh(x)+C int cosh(x)dx=sinh(x)+C int tanh(x)dx=ln(cosh(x))+C int coth(x)dx=ln|sinh(x)|+C int sech(x)dx=arctan|sinh(x)|+C int csch(x)dx=ln|tanh(1/2 x)|+C int sech^2(x)dx=tanh(x)+C int csch^2(x)dx=-coth(x)+C int sech(x) tanh(x)dx=-s ech(x)+C int csch(x) coth(x)dx=-c sch(x)+C Forms Involving sqrt(a^2+x^2), a>0 int sqrt(a^2+x^2)dx=x/2sqrt(a^2+x^2)+(a^2)/2ln(x+sqrt(a^2+x^2))+C int x^2sqrt(a^2+x^2)dx=x/8(a^2+2x^2)sqrt(a^2+x^2)-(a^4)/8ln(x+sqrt(a^2+x^2))+C int (sqrt(a^2+x^2))/x dx=sqrt(a^2+x^2)-a ln|(a+sqrt(a^2+x^2))/x|+C int (sqrt(a^2+x^2))/(x^2) dx=-(sqrt(a^2+x^2))/x+ln(x+sqrt(a^2+x^2))+C int (dx)/(sqrt(a^2+x^2))=ln(x+sqrt(a^2+x^2))+C int (x^2dx)/(sqrt(a^2+x^2))=x/2 sqrt(a^2+x^2)-(a^2)/2ln(x+sqrt(a^2+x^2))+C int (dx)/(xsqrt(a^2+x^2))=-1/a ln|(sqrt(a^2+x^2)+a)/x|+C int (dx)/(x^2 sqrt(a^2+x^2))=-(sqrt(a^2+x^2))/(a^2x)+C int (dx)/((a^2+x^2)^(3/2))=x/(a^2 sqrt(a^2+x^2))+C Forms Involving sqrt(a^2-x^2), a>0 int sqrt(a^2-x^2)dx=x/2sqrt(a^2-x^2)+(a^2)/2 arcsin(x/a)+C int x^2sqrt(a^2-x^2)dx=x/8(2x^2-a^2)sqrt(a^2-x^2)+(a^4)/8 arcsin(x/a)+C int (sqrt(a^2-x^2))/x =sqrt(a^2-x^2)-a ln|(a+sqrt(a^2-x^2))/x|+C int (sqrt(a^2-x^2))/(x^2)dx=-1/x sqrt(a^2-x^2)-arcsin(x/a)+C int (x^2)/sqrt(a^2-x^2)dx=-x/2 sqrt(a^2-x^2)+(a^2)/2 arcsin(x/a)+C int (dx)/(xsqrt(a^2-x^2))=-1/a ln|(a+sqrt(a^2-x^2))/x|+C int (dx)/(x^2 sqrt(a^2-x^2))=-1/(a^2 x)sqrt(a^2-x^2)+C int (a^2-x^2)^(3/2)dx=-x/8(2x^2-5a^2)sqrt(a^2-x^2)+(3a^4)/8 arcsin(x/a)+C int (dx)/((a^2-x^2)^(3/2))=x/(a^2 sqrt(a^2-x^2))+C Form Involving sqrt(x^2-a^2), a>0 int sqrt(x^2-a^2)dx=x/2 sqrt(x^2-a^2)-(a^2)/2 ln|x+sqrt(x^2-a^2)|+C int x^2 sqrt(x^2-a^2)dx=x/8(2x^2-a^2)sqrt(x^2-a^2)-(a^4)/8 ln|x+sqrt(x^2-a^2)|+C int (sqrt(x^2-a^2))/x dx=sqrt(x^2-a^2)-a*arccos(a/(|u|))+C int (sqrt(x^2-a^2))/(x^2) dx=-(sqrt(x^2-a^2))/x+ln|x+sqrt(x^2-a^2)|+C int (dx)/sqrt(x^2-a^2)=ln|x+sqrt(x^2-a^2)|+C int (x^2)/(sqrt(x^2-a^2)) dx=x/2 sqrt(x^2-a^2)+(a^2)/2 ln|x+sqrt(x^2-a^2)|+C int (dx)/(x^2 sqrt(x^2-a^2))=(sqrt(x^2-a^2))/(a^2 x)+C int (dx)/((x^2-a^2)^(3/2))=-x/(a^2 sqrt(x^2-a^2))+C Forms Involving a+bx int x/(a+bx)dx=1/(b^2)(a+bx-aln|a+bx|)+C int (x^2)/(a+bx)dx=1/(2b^3)((a+bx)^2-4a(a+bx)+2a^2 ln|a+bx|)+C int (dx)/(x(a+bx))=1/a ln|x/(a+bx)|+C int (dx)/(x^2 (a+bx))=-1/(ax)+b/(a^2)ln|(a+bx)/x|+C int x/((a+bx)^2)dx=a/(b^2(a+bx))+1/(b^2)ln|a+bx|+C int (dx)/(x(a+bx)^2)=1/(a(a+bx))-1/(a^2) ln|(a+bx)/x|+C int (x^2)/((a+bx)^2)dx=1/(b^3)(a+bx-(a^2)/(a+bx)-2a ln|a+bx|)+C int xsqrt(a+bx)dx=2/(15b^2)(3bx-2a)(a+bx)^(3/2)+C int x/(sqrt(a+bx))dx=2/(3b^2)(bx-2a)sqrt(a+bx)+C int (x^2)/(sqrt(a+bx))dx=2/(15b^3)(8a^2+3b^2x^2-4abx)sqrt(a+bx)+C int (dx)/(xsqrt(a+bx))={(1/(sqrt(a))ln|(sqrt(a+bx)-sqrt(a))/(sqrt(a+bx)+sqrt(a))|+C if a>0),(2/(sqrt(-a))arctan(sqrt((a+bx)/(-a)))+C if a<0):} int (sqrt(a+bx))/x dx=2sqrt(a+bx)+a int (dx)/(xsqrt(a+bx)) int (sqrt(a+bx))/(x^2)dx=-(sqrt(a+bx))/x+b/2 int (dx)/(xsqrt(a+bx)) int x^n sqrt(a+bx)dx=2/(b(2n+3))(x^n (a+bx)^(3/2)-n a int x^(n-1)sqrt(a+bx)dx) int (x^n)/(sqrt(a+bx))dx=(2x^n sqrt(a+bx))/(b(2n+1))-(2na)/(b(2n+1)) int (x^(n-1))/(sqrt(a+bx))dx int (dx)/(x^n sqrt(a+bx))=-(sqrt(a+bx))/(a(n-1)x^(n-1))-(b(2n-3))/(2a(n-1)) int (dx)/(x^(n-1) sqrt(a+bx)) Forms Involving sqrt(2ax-x^2), a>0 int sqrt(2ax-x^2)dx=(x-a)/2 sqrt(2ax-x^2)+(a^2)/2 arccos ((a-x)/a)+C int xsqrt(2ax-x^2)dx=(2x^2-ax-3a^2)/6 sqrt(2ax-x^2)+(a^3)/2 arccos((a-x)/a)+C int (sqrt(2ax-x^2))/x dx=sqrt(2ax-x^2)+a arccos((a-x)/a)+C int (sqrt(2ax-x^2))/(x^2)dx=-(2sqrt(2ax-x^2))/x-arccos((a-x)/a)+C int (dx)/(sqrt(2ax-x^2))=arccos((a-x)/a)+C int x/(sqrt(2ax-x^2)) dx=-sqrt(2ax-x^2)+a arccos((a-x)/a)+C int (x^2)/(sqrt(2ax-x^2))dx=-(x+3a)/2 sqrt(2ax-x^2)+(3a^2)/2 arccos((a-x)/a)+C int (dx)/(xsqrt(2ax-x^2))=-(sqrt(2ax-x^2))/(ax)+C