Calculus: Rules of Integration

Calculus Integration Rules
Following are the Rules of Integration where a, b, c, n are some constants and u=f(x);v=g(x) and w=h(x)
General Integration Rules
1.Integral [a dx]= ax +c
2. Integral [a f(x) dx]= a Integral [f(x) dx]
3.Integral [x^n dx]= x^(n+1)/(n+1) +c
4.Integral [f(x)+g(x)+h(x)]dx = Integral[f(x)dx]+ Integral[g(x)dx]+ Integral[h(x)dx]
5. Integral[f(x)-g(x)-h(x)]dx = Integral[f(x)dx] – Integral[g(x)dx] – Integral[h(x)dx]
6. Integration by parts: Integral [u dv] = uv – Integral [v du]
7.Integral[F(u)dx] = Integral [F(u)/u’] du
8. Integral[1/x dx] =ln|x| +c
9. Integral [1/(x^2+a^2)]dx =1/a tan^-1[x/a] +c
10.Integral [1/(x^2-a^2)]dx = [1/2a ]ln |x-a/x+a| +c

Integral Rules of Exponential Functions
1.Integral [e^x dx]= e^x +c
2. Integral [a^x dx]= a^x/ln a +c
3.Integral [ln x dx]= x(ln x -1) +c
4.Integral[log base a of x]dx= (x/ln a)(ln x -1) +c
5. Integral[x e^(ax)]dx= [e^(ax)/a^2](ax-1) +c
6. Integral[e^(ax)/x] dx = ln|x| + summation(i=1 to infinity) [(ax)^i/i.i!] +c
7.Integral[x^2 e^(ax)]dx= e^(ax)[(x^2/a – 2x/a^2 + 2/a^3)] +c
8. Integral[x^n e^(ax)]dx = (1/a)x^n e^(ax) – (n/a)Integral [x^(n-1) e^(ax)]dx
9.Integral[e^(ax)/x^n] dx = [1/(n-1)][- e^(ax)/x^(n-1) + a Integral e^(ax)/x^(n-1)]dx
10.Integral[x^n ln x] dx= [x^(n+1)]/(n+1)^2 [(n+1)lnx – 1] + c

Integration Rules of Trigonometric Functions
1.Integral [sin x dx]=  - cos x +c
2. Integral [cos x dx]= sin x +c
3. Integral [tan x dx]= ln |sec x|+ c
4. Integral [cot x dx]= ln|sin x| + c
5. Integral [sec^2(x) dx]= tan x +c
6. Integral [csc^2(x) dx]= - cot x +c
7. Integral [tan^2(x) dx]= tan x – x +c
8. Integral [cot^2(x) dx]= cot x – x +c
9.Integral [sec x tan x dx]= sec x +c
10. Integral [csc x cot x dx] = - csc x +c
11. Integral [sec x dx]= ln |sec x +tan x| +c
12.Integral [cos^2(x) dx] = x/2 + ¼(sin 2x) +c
13. Integral[sin ^n(x) dx]= (-1/n)sin^(n-1) x cos x + (n-1)/n .Integral [sin^(n-2) x dx]
14. Integral [cos^n(x)dx]= (1/n)cos^(n-1) x sin x + (n-1)/n. Integral[cos^(n-2) dx]

Integration Rules of Hyperbolic Functions
1.Integral[sinh x dx] = cosh x +c
2.Integral[cosh x dx]= sinh x +c
3. Integral[tanh x dx]= ln cosh x +c
4. Integral[coth x dx]= ln |sinh x| + c
5. Integral[sech x dx] = sin^-1[tanh x] +c
6. Integral[csch x dx] = ln tanh (x/2) +c
7. Integral[sinh^2(x)dx] = (sinh 2x)/4 – (x/2) + c
8. Integral[cosh^2(x)dx]= (sinh 2x)/4 + (x/2) +c
9. Integral[sech^2 (x) dx]= tanh x +c
10. Integral[csh^2(x)dx]= -coth x +c
11. Integral[tanh^2(x) dx]= x – tanh x +c
12. Integral[coth^2(x)dx]= x – coth x +c
13.Integral[sechx tanh x] dx =  - sech x +c
14. Integral[csch x coth x]dx = -csch x +c