Integrali i Pacaktuar. Tabela dhe vetitë

22.1

F(x) quhet primitive e funksionit f(x) nese F'(x)=f(x). Bashkesina e primitivave F(x)+c te funksionit f(x) quhet integral i pacaktuar i funksionit f(x).

 
 
 
Tabela e integraleve

1) \int x^{\alpha }dx=\frac{x^{\alpha +1}}{\alpha +1}+c     

2)\int dx=x+c

3)\int \frac{dx}{\sqrt{x}}=2\sqrt{x}+c

4)\int \frac{dx}{x}=ln\left | x \right |+c

5)\int a^{x}dx=\frac{a^{x}}{lna}+c

6)\int e^{x}dx=e^{x}+c

7)\int sinxdx=-cosx+c

8)\int cosdx=sinx+c

9)\int \frac{dx}{cos^{2}x}=tgx+c

10)\int \frac{dx}{sin^{2}x}=-cotgx+c

11)\int \frac{dx}{x^{2}-a^{2}}=\frac{1}{2a}ln\left | \frac{x-a}{x+a} \right |+c

12)\int \frac{dx}{\sqrt{x^{2}+a}}=ln\left | x+\sqrt{x+a} \right |+c

13)\int \frac{dx}{1+x^{2}}=artgx+c=-arcctgx+c

14)\int \frac{dx}{a^{2}+x^{2}}=\frac{1}{a}artg\frac{x}{a}+c , a\neq 0

15)\int \frac{dx}{\sqrt{1-x^{2}}}=arcsinx+c=-arccosx+c

16)\int \frac{dx}{\sqrt{a^{2}-x^{2}}}=arcsin\frac{x}{a}+c, a>0

17)\int tgxdx=-ln\left | cosx \right |+c

18)\int cotgxdx=ln\left | sinx \right |+c

19)\int shxdx=chx+c

20)\int chxdx=shx+c

21)\int \frac{dx}{ch^{2}x}=thx+c

22)\int \frac{dx}{sh^{2}x}=-cthx+c

Vetite e integralit te pacaktuar:

1)\left ( \int f(x)dx \right )'=f(x)

2) d\left ( \int f(x)dx \right )=f(x)dx

3)\int F'(x)dx+F(x)+c

4)\int dF(x)=F(x)+c

5)\int \left [ f(x)\pm g(x) \right ]dx=\int f(x)dx\pm \int g(x)dx

6)\int kf(x)dx=k\int f(x)dx