Integrais imediatas

1) du = u + C

 

2) kdu = ku + C

Ex.: 8du = 8 du = 8u + c

 

3) (k+l+m)du = ku+lu+mu+C

 

4) u^mdu = u^m+1/m+1 + C

Ex.: 8x⁹dx = 8 x⁹dx = 8(x¹⁰/10) + C

 

5) cos u du = sen u + C

Ex 1.: cos 3x dx = 1/3 cos 3x 3dx = (1/3)sen 3x + C

Ex 2.: cos nx dx = 1/n cos nx ndx = (1/n)sen nx + C

 

6) sen u du = -cos u+C

Ex 1.: sen 4x dx = (1/4) (-cos 4x 4dx) = (-1/4)cos 4x + C

Ex 2.: sen nx dx = (1/n) (-cos nx ndx) = (-1/n)cos nx + C

 

7) tg u du = -ln | cos u | + C ou ln | sec u| + C

Ex 1.: tg 2x dx = (1/2) tg 2x 2dx = -(1/2)ln | cos 2x | + C ou (1/2)ln | sec 2x| + C

Ex 2.: tg nx dx = (1/n) tg nx ndx = -(1/n)ln | cos nx | + C ou (1/n)ln | sec nx| + C

 

8) sec u du = ln | sec u+tg u| + C

Ex 1.: sec 10x dx = (1/10) sec 10x 10dx = (1/10) ln | sec 10x + tg 10x| + C

Ex 2.: sec nx dx = (1/n) sec nx ndx = (1/n) ln | sec nx + tg nx| + C

 

9) cossec u cotg u du = -cossec u+C

Ex 1.: cossec 5x cotg 5x dx = (1/5)cossec 5x cotg 5x 5dx = -(1/5)cossec 5x + C

Ex 2.: cossec Kx cotg Kx dx = (1/K)cossec Kx cotg Kx Kdx = -(1/K)cossec Kx + C

 

10) a^udu = a ^u/lna + C

Ex 1.: a^15xdx = (1/15)a^15x15dx = (1/15) a^15x/lna + C

Ex 2.: a^Kxdx = (1/K)a^KxKdx = (1/K) a^Kx/lna + C

 

11) e^udu = e^u + C

Ex 1.: e^2xdx = (1/2)e^2x2dx =e^2x + C

Ex 2.: e^pxdx = (1/p)e^pxpdx =e^px + C

 

12) du/u = ln|u|+ C

Ex 1.: dx/|x+3| = ln|x+3|+ C

Ex 2.: sec²x dx / tg x = ln|tg x|+ C

 

13) sec u du = ln|sec u+tg u| + C

Ex 1.: sec 3s ds = (1/3)sec 3s 3ds = (1/3)ln|sec 3s + tg 3s| + C

Ex 2.: sec Ks ds = (1/K)sec Ks Kds = (1/K)ln|sec Ks + tg Ks| + C

 

14) du/(a² + u²) = (1/a) arc tg (u/a) + C

Ex.: (cosh x dx) /(81 + senh²x) = (1/9) arc tg (senh x)/a] + C

 

15) du/(a²-u²)^1/2 = arc sen 1/a + C

Ex.: 2xdx/(121²-x²)^1/2 = arc sen 1/11 + C