Integrales

Topic 5: INTEGRATION
• Indefinite integrals • Some important integrals • Integration by parts • Change of variable. Integration by substitution. • Integration of rational functions

INDEFINITEINTEGRAL
• Each function F(x): (a,b) →ℜ that verifies F’(x) = f(x) ∀x∈(a,b) is called an indefinite (Primitive) integral of f(x) on (a,b). Fundamental Theorem of the Integral Calculus Given twoindefinite integrals of the same function f, F1(x) and F2(x), throughout an interval (a,b), they differ by a constant: F1(x) = F2(x)+C ∀x∈(a,b) Definition • The set of all the indefinite integral functionsof a function f(x) (F(x)+C) is called the INDEFINITE Integral of f(x) and it is writen as

∫ f(x)dx = F(x) + C

PROPERTIES OF THE INDEFINITE INTEGRALS

d 1. ∫ f ( x)dx = f ( x), d ∫ f ( x)dx = f( x) dx d 2. ∫ F ( x)dx = F ( x) + C, ∫ dF ( x)dx = F ( x) + C dx 3. ∫ [ f ( x) ± g ( x) ± ... ± h( x) ]dx = ∫ f ( x)dx ± ∫ g ( x)dx ± ... ± ∫ h( x)dx 4. ∫ kf ( x)dx = k ∫ f ( x)dx

INTEGRALSREDUCIBLES TO INMEDIATES
f(x)p+1 ∫ f(x)p f'(x)dx = p + 1 + C, p ≠ −1;

e f(x )f'(x)dx = e f(x ) + C; ∫

f'(x) ∫ f(x) dx = ln f(x) + C;
f'(x)dx ∫ cos2 f(x) = tgf(x) + C;

∫ f'(x)senf(x)dx = − cosf(x) + C ∫ f'(x)cos f(x)dx = senf(x) + C

∫a

f(x )

a f(x ) f'(x)dx = + C; ln a

∫

f'(x) 2 dx = arcsen f(x) + C 1− f(x)

f'(x) ∫ 1+ f(x)2 dx = arctag f(x) + C;

∫

−f'(x) 2 dx =arccos f(x) + C 1− f(x)

SOME INDEFINITE INTEGRALS (Inmediate Integrals)

∫ kdx = kx + C
x p+1 p + C, p ≠ −1 x dx = ∫ p +1 1 ∫ x dx = lnx + C
e x dx = e x + C ∫

∫ senxdx = − cos x + C ∫ cos xdx =sen x + C

∫ ∫

1 2 dx = arcsen x + C 1− x −1 2 dx = arccos x + C 1− x

ax a x dx = +C ∫ lna

1 ∫ 1+ x 2 dx = arctagx + C

INTEGRATION BY PARTS Given two functions u(x) and v(x)derivables, with continuous first derivative, then:

∫ u(x)v'(x)dx = u(x)v(x) − ∫ v(x)u'(x)dx
Writing u=u(x) and v=v(x), we have du=u’(x)dx and dv=v’(x)dx. From this, (1) can be written as

∫ udv = u.v −...