PROBLEMARIO RESUMEN GUIAS PUBLICADAS POR EL DEPARTAMENTO DE MATEMATICAS PURAS Y APLICADAS DE LA UNIVERSIDAD SIMON BOLIVAR

TRIMESTRE: ENERO – MARZO 2008.

DISPONIBILIDAD http://ma.usb.ve/cursos/

La guías a continuación corresponde a la semana 1,2,3,4,5,6,7,8 tiene las soluciones de las guías 1,2,4,5,6,7.

Sírvase de ayuda para practicar matemáticas 2.

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√

x+

√1 x

2

x1/3 + x−1/3 +

1 1+4x2

2

−π cos( πx ) − sec(x) tan(x)2 2

− csc(5t) cot(5t) + 32 t−5/2 + t2 + 5t − 8.

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  2x − 2, 5, f (x) =  2 3x − 2x

'C

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√

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³ n ´n e

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sen(x) cos(x) + cos(x) 1 + sen(x)

9 *"'+( B6%(52"5(C &$*(!( D B

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">

?0

R

D0 B"'("-=, -,' '%)(%"/3"' %/3")#,-"' (3%-%9,/*& $&.+-"3,$%8/ *" $(,*#,*&' > !> "> #>

R

√ dx dx x2 +4x+5

R√

R√

R

7 '()"#"/$%,4 (x + 2)2 + 1 = x2 + 4x + 5 : x + 2 = tan(t)0

−x2 + x + 1dx7 '()"#"/$%,4

5 4

− (x − 12 )2 = −x2 + x + 1 : x −

t2 − 6tdt7 '()"#"/$%,4 t − 3 = 3 sec(x)0

2x−1 dx x2 −6x+13

2x−1 7 '()"#"/$%,4 (x−3) = (2x−6)+5 0 +4 (x−3) +4 2

2

E0 #> $> %> &> '>

R

sen(4y) cos(5y)dy 0

R

sen4 (3t) cos4 (3t)dt0

R

tan−3 (x) sec4 (x)dx0

R

tan4 (x)dx0

csc3 (y)dy 0 p R π/2 sen3 (z) cos(z)dz 0 π/4 R 3 sen(z) dz 0 cos2 (z)+cos(z)−2 R dt R

7 '()"#"/$%,4 1 + cos2 (t) = 2 cos2 (x) + sen2 (x)7 sec (t) 1 2 cos (x) + sen2 (x) = cos2 (t)(2 + tan2 (t)) : 1+cos = 2+tan 0 (t) (t) R ( > x sen3 (x) cos(x)dx7 '()"#"/$%,4 (3%-%$" %/3")#,$%8/ +&# +,#3"'0 1+cos2 (t) 2

2

2

2

1 2

=

√

5 2

sen(t)0

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dx cot(2x)(1−cos(2x))

R

dx x3 +1

R

dx x(3−ln(x))(1−ln(x))

R

(x3 −8x2 −1)dx (x+3)(x−2)(x2 +1)

R

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1 1

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(x+1)dx (x−3)2

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(3x2 +2x−2)dx x3 −1

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1

1 1

1 1

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P3

i=1

A(Ri ) = 68

3 b−a = , ∀i, 1 ≤ i ≤ n. n n

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2.5

2

1.5

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P4

i=1

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1

1.5

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1.5

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0.5

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1.5

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0.5

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k X

A(Ri ) =

i=1

k X 3 i=1

k

3i k

0.5

1

1.5

2

%

f (xi ) =

k k X 9 X 9 k(k + 1) 9i = i= 2 . 2 2 k k k 2 i=1 i=1

= 4, 5+ &'()*$ l´ımk→∞ 92 k(k+1) k2 , f (x) = 3x2 + x + 1- a = −1 % b = 1+

*+,-./012

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"D

sen(x) dx 2−sen2 (x)

R

sen(4x) dx cos(2x) cos(x)

R 2 sen(2x) cos(2x) R = dx = 2 sen(2x) dx cos(2x) cos(x) R 2 sen(x) cos(x) R cos(x) = 2 dx = 4 sen(x)dx cos(x) = −4 cos(x) + C.

cos3 (3x) sen(3x)dx2

!"#$%&'(

R

R cos3 (3x) sen(3x)dx = R cos2 (3x) cos(3x) sen(3x)dx = (1 − sen2 )(3x) cos(3x) sen(3x)dx

B",-%F,/*& "- $,.4%& *" C,#%,4-" u = 1 − sen2 (3x) 0 du = −6 sen(3x) cos(3x)dx9 &45"/J ".&' G(" Z

Z 1 −u2 (1 − sen )(3x) cos(3x) sen(3x)dx = − udu = +C 6 2 −(1 − sen2 (3x))2 + C. = 2 2

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!

0 −2

*+,-./012

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4

w−2+1 4 −1 4 −1 −1 1 3 dw = |1 = |1 = − = . 2 w −2 + 1 w 4 1 4

1

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(x − 2|x|)dx%

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2

−1

#!

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  3x () x < 0

−x () x ≥ 0 Z 0 Z (x − 2|x|)dx = 3xdx + 

2

0

−1

3x2 0 −x2 2 −7 |−1 + | = . 2 2 0 2

−xdx =

sen2 (3x) cos(3x)dx%

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p

3− | t |dt = 2

√

0

3 − tdt

5#',$/# +- ",'0)# /+ 1,2),0-+ u = 3 − t @du = −dt!3 Z

0

3

√

3 − tdt = −

Z

3

0

√

udu =

Z

0

3

√

udu

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Z

−3

)

Rπ

√

udu =

0

√ 4u3/2 3 |0 = 4 3. 3

(x5 + | sen(x) |)dx*

*+,-./012 −π

3

Z p 3− | t |dt = 2

+,-, f (x) =| sen(x) |=

  − sen(x) "& −π ≤ x ≤ 0

"& 0 ≤ x ≤ π +./'/-$01$( f $" 20/ 320%&40 5/' 6 x 20/ 320%&,0 &-5/'* !01,0%$"( Z

)

R π/3

sen(x)

5

π 5

−π

!



π

Z

(x + | sen(x) |)dx =

5

x dx +

Z

π

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−π

sen5 (θ)dθ*

*+,-./012 −π/3

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*+,-./012

Z

1

f (t)dt =

−2

int0−2 g(t)dt

+

Z

1

h(t)dt

0

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−2

f (t)dt = −

R0

−2

(t2 − 2t)dt −

R1

tdt = 2 −

Z

4

0

1 2

=

3 2

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Z

4

f (x)dx =

0

Z

1

dx +

0

Z

2

xdx +

1

2

9 (4 − x)dx = . 2

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73'5 Pn

k=1

8 n

¢ P ( nk=1 k) − 8 =

f (xk−1 )∆x =

2 n

¡ 11n +

=

2 n

× (15n − 4)

=

30n−4 . n

2 n

³

11n +

8 n(n+1) n 2

´ −8

8!*+/5 l´ım

n→∞

n X

30n − 4 = 30 n→∞ n

f (xk−1 )∆x = l´ım

k=1

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2

4

(4x + 3)dx = l´ım

n→∞

n X

f (xk−1 )∆x = 30.

k=1

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6

7

R9

(3 − 0

8

√

9

x)xdx =

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√

6 x = 2 y, x = 0, y = 41

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2.5

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3

3.5

4

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5

10

15

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20

R 27 0

25

30

x(9 − x2/3 )dx =

6561π 4

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6 y = 4x, y = 4x2 1

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0

0.2

0.4

0.6

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1

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2π 3

1 2"/& 3).#4#$#(&(5

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1

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−0.5

0

0.5

1

A(x)dx! "#$ A(x) = l2 (x) %#&'()*( +,( -. )(/01$ 2( -. &.'( 2(- '#-02# (' '03(4)0". "#$ )('5("4# .- (6( y 78 9- -.2# 2(- ",.2).2# R($ :,$"01$ 2( -. *.)0.&-( x ('4. 2.2# 5#) 1 16 l(x) = (1 − x4 ) − (1 − x2 ) = x2 − x4 8 ;'0! V = 2 0 (x2 − x4 )2 dx = 315 8 V =2

R1 0

R

y

Vn = 2 π R · H · ∆ y

H

∆y

x

TUBOS

Vn = 2 π R · H · ∆ x

H

y ∆x

R

x

V = π R2 · ∆x

∆x

R

DISCOS

V = π R2 · ∆y

y

R

x

∆y

y

V = π ( R2 – r2 ) · ∆x

∆x

r

R

x

ARANDELAS

V = π ( R2 – r2 ) · ∆y

y

r

R

x

∆y

Recinto generador

R

x R

Sólido de revolución generado

H

H

y=f(x)

y

y

x

Sólido de revolución generado por un recinto plano al girar alrededor del eje OY

y0 = f ( x0 )

H

y

x0

∆x

x

Por tubos: V =

H

y

x =0

³

x

2 π x ⋅ [ H − f ( x )] dx

x= R

x0

∆x

Proyección sobre el eje OX:

V0 = 2 π x0 [ H – f ( x0 )] ∆x

x0

H – f ( x0 )

∆x

y0

H

x0 = f -1 ( y0 ) R

∆y

y=f(x)

x

∆y

Por discos: V =

H

y

y =0

³

y= H

2

x

−1 𠪫 f ( y )º» dy ¬ ¼

R

Proyección sobre el eje OY:

∆y

V0 = π [ f –1( y0 ) ] 2· ∆y

x0

Recinto generador

R

x R

Sólido de revolución generado

H

H

y=f(x)

y

y

x

Sólido de revolución generado por un recinto plano al girar alrededor del eje OX

y0 = f ( x0 )

H

y

x0

∆x

R

x

Por arandelas: V =

H

y

x =0

³

x=R

x0

∆x

2 π ⋅ ª« H − [ f ( x )] 2 º» dx ¬ ¼

R

x

Proyección sobre el eje OX:

y0 = f ( x0 )

H

x

V0 = π· [ H2 – f2(x0) ] · ∆x

xx00

∆x

y0

x0 = f -1 ( y0 ) R

∆y x

Por tubos: V =

H

H

y=f(x)

y

y

y =0

³

y= H

2πy⋅ f

R

−1

x

( y ) dy

Proyección sobre el eje OY:

x0

V0 = 2 π y0 · f-1 ( y0 ) · ∆y

y0

∆y

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!"#$%$%&' '()"#%*&' +,#, -, '".,/, 6 01& 72 3(4#" "- '%)(%"/5" .,5"#%,-6 37-$(-& *" 8&-(."/"' *" '9-%*&' *" #"8&-($%9/: ;(/$%&/"' "2 ?,--" -,' *"#%8,*,' *" -,' '%)(%"/5"' ;(/$%&/"'6 @

z = x3 ln(x2 ) + (log7 (πx + e))5 2

!"#$%&'(

5π (log7 (πx + e))4 dz = 3x2 ln(x2 ) + 2x2 + . dx ln(7)(πx + e) √

!@ y = ex2 + e

√

!"#$%&'(

" @ y = 32x

2 −3x

!"#$%&'(

x2

2

2

dy x exp(x2 ) =p + exp(x). dx exp(x2 )

dy 2 = ln(3)(4x − 3)32x −3x . dx

#@ y =

p log10 (3x2 −x )2

!"#$%&'(

$@ y = xπ+1 + (1 + π)x 2

dy ln(3)(2x − 1) p . = dx 2 ln(10) log10 (3x2 −x )

!"#$%&'(

¡ ¢ dy = (π + 1) xpi + ln(π + 1)(π + 1)x−1 . dx

% @ y = xx $&/ x > 02

!"#$%&'(

dy = xx (1 + ln(x)). dx

!"#$ % &'"%&'"()'*

!"###$

! y = g(x)f (x) " #$%&'() *$+ g(x) , f (x) #&' -./+0+'1.)23+# , g(x) +# #.+4%0+ %&#.5.6)7

*+,-./012

8. g (x) = 0 ⇔ g(x) = C 1&' C > 0" +'5&'1+# ′

 ³ ′   g(x)f (x) f (x) ln(g(x)) +

dy = dx  

¡

f (x) g ′ (x)

´

¢ ′ f (x) ln(C) C f (x)

′

8.

g (x) 6= 0 ′

g (x) = 0

!! ex+y = 4 + x + y 9-+0.6)-) .4%3.1.5)!7

*+,-./012

8$%&'()4&# *$+ y -+%+'-+ -+ x" -+0.6)'-& )42&# 3)-&# -+ 3) +1$)1.:' 1&' 0+#%+15& ) x7 ;# -+1.0" d d ¡ x+y ¢ e (4 + x + y), = dx dx

&25+'+4&# *$+ ¡ ′¢ 1 + y ex+y

= 1+y

′

ex+y + y ex+y = 1 + y

′

′

ex+y − 1 ⇒

dy dx

′

= y (1 − ex+y )

= −1.

"! y = coth( arctanh(x))7

*+,-./012

dy csch2 ( arctanh(x)) =− . dx 1 − x2

#! y = ln( arccosh(x))7

*+,-./012

1 dy . =√ dx 1 − x2 arccosh(x)

$! y = 5 senh2 (x) + x2 cosh(3x) − x arcsenh(x3 )7

*+,-./012

dy 3x3 . = 10 senh(x) cosh(x) + 2x cosh(3x) + 3x2 senh(3x) − arcsenh(x3 ) − √ dx x2 + 1

7 + y 7 ;'1$+'50+ +3 6&3$4+' -+3 #:3.-& -+ 0+6&3$1.:' 0+#$35)'5+ 96+0 +3 (0?@1&!7 2

!"#$ % &'"%&'"()'*

!"###$

Gráfica de y=exp(−x2) 1.2 1 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −3

−2

−1

0

1

2

3

*+,-./012

!" #$%&'( $)*+",-.,) /) $(,-$ "- $)#01. -'(,-/- -"$)/)/($ /)" )2) y *) 3+)*,$- ). "- *0#+0).,) &#+$-

1 0.8 0.6 0.4 0.2 0 2 1

2 1

0

0

−1

−1 −2

−2

40 +,0"05-3(* )" 36,(/( /) '-*'-$(.)*7 V = 2π 02 x exp(−x2 )dx8 9)-"05-./( )" '-3:0( /) R ;-$0-:") u = x2 7 (:,).)3(* x = ln(5)8

*+,-./012

?- $)#01. -'(,-/- =($ y = cosh(2x), y = 0, x = − ln(5) > x = ln(5) y *) 3+)*,$- ). "*0#+0).,) &#+$14

12

10

8

6

4

2

0 −2

−1.5

−1

−0.5

0

0.5

1

1.5

2

!"#$ % &'"%&'"()'*

!"###$

!A=2

R ln(5) 0

ln(5)

cosh(2x)dx = senh(2x)|0

= 2, 4"

" #$%%& %$' '()*(&+,&' (+,&)-$%&' .

x2x dx" 2

R

*+,-./012 2

2x −1 + C. x2 dx = ln(2)

Z

!.

2 ln(x) dx x

R

"

*+,-./012 2 ln(x) dx = ln2 (x) + C. x

Z

".

R1

(103x + 10−3x ) dx"

*+,-./012 0

Z

1

0

#.

R

(x + 3)ex

2 +6x

*+,-./012

¡ 3x ¢ 10 + 10−3x dx =

R1

2 +6x

(x + 3)ex

e2x+3 dx" 1

e2x+3 dx =

0

R

6v+9 dv 3v 2 +9v

*+,-./012

R

6v + 9 dv = ln |3v 2 + 9v| + C. 3v 2 + 9v

cot(θ)dθ"

*+,-./012

Z

'.

¢ 1¡ 5 e − e3 . 2

" Z

&.

1 2 dx = ex +6x + C. 2

*+,-./012 0

Z

%.

¡ 3 ¢ 1 10 − 10−3 . 3 ln(10)

dx" Z

$.

x2

R

cot(θ)dθ = ln | sen(θ)| + C.

sec(u) csc(u)du"

(u)+cos (x) *+,-./012 /$01 2*& sec(u) csc(u) = sen(u)1cos(u) = sensen(u) = tan(u) + cot(u) cos(u)

Z

sec(u) csc(u)du =

2

Z

tan(u)du +

Z

2

cot(u)du = − ln | cos(u)| + ln | sen(u)| + C.

!"#$ % &'"%&'"()'*

!"###$R !

x coth(x2 ) ln(senh(x2 ))dx"

*+,-./012

#$%&'(%)*+ $& ,%-.'+ *$ /%0'%.&$ u = ln(senh(x2 ))1 2$ +.3'$)$ 45$ Z

1 x coth(x ) ln(senh(x ))dx = 2 2

2

Z

u2 du =

u3 + C. 6

6$/+&/$-+2 $& ,%-.'+ *$ /%0'%.&$ 0$%&'(%*+ Z !

!

R

senh(2z 1/4 ) √ dz 4 3 z

*+,-./012

ln3 (senh(x2 )) x coth(x ) ln(senh(x ))dx = + C. 6 2

2

"

#$%&'(%)*+ $& ,%-.'+ *$ /%0'%.&$ u = 2z 1/4 1 2$ +.3'$)$ 45$ Z

senh(2z 1/4 ) √ dz = 2 4 z3

Z

senh(u)du = 2 cosh(u) + C.

6$/+&/$-+2 $& ,%-.'+ *$ /%0'%.&$ 0$%&'(%*+ Z

senh(2z 1/4 ) √ dz = 2 cosh(2z 1/4 ) + C. 4 z3

" 7+)2'*$0$ &% 0$8'9) R 45$ 2$ -5$230% $) &% 2'85'$)3$ :850%" ;+0-5&$ 5)% ')3$80%& 3+*+ 45$ 2$ ')*',%"

"

! ?& $@$ x A%0%)*$&%2!"

*+,-./012

R = f (x) B r = g(x)1 $)3+),$2 V = π #

! ?& $@$ y A,%2,%0+)$2!"

*+,-./012 Rb V = 2π

a

x (f (x) − g(x)) dx"

Rb a

(f 2 (x) − g 2 (x)) dx"

!"#$ % &'"%&'"()'*

!"###$

! "# $%&'# x = a (&#)&#$*+%)!,

*+,-./012 Rb V = 2π

a

(x − a) (f (x) − g(x)) dx,

! ! "# $%&'# x = b (&#)&#$*+%)!,

*+,-./012 Rb V = 2π

a

(b − x) (f (x) − g(x)) dx,

, -*+)./%$% 0# $%1.2+ R 34% )% 54%)'$# %+ 0# ).14.%+'% 614$#, 7*$540% 4+# .+'%1$#0 8#$# %0 9*045%+ /%0 )20./* 34% )% 1%+%$#&4#+/* )% :#&% 1.$#$ R #0$%/%/*$ /% 0# $%&'# /#/#; 4'.0.&% %0 5% x (&#)&#$*+%)!,

*+,-./012 Rd V = 2π

c

Rd c

(f 2 (y) − g 2 (y)) dy ,

y (f (y) − g(y)) dy ,

! "# $%&'# y = 3 (&#)&#$*+%)!,

*+,-./012 Rd V = 2π

c

(3 − y) (f (y) − g(y)) dy ,

!"#$%$%&' &($%&)*+"'

@, "# .+'%+)./#/ /%0 )*+./* )% 5./% %+ /%&.A%0%); %+ :*+*$ /% B0%>#+/$* C$#:#5 D%00 (EFG@H EIJJ!; .+9%+'*$ /%0 '%0/'8$() 1% ?&+'(+"5 1+#% @$% -)') n 5')"1% -/1%8/. )-'/A+8)' n! = 1 × 2 × · · · × n -/' n! ≈

√

2πn

³ n ´n e

B)(#$(% 10! 1% 8)"%') %A)#&)4 ($%5/ 1% >/'8) )-'/A+8)1) 8%1+)"&% () >/'8$() )"&%'+/'3

*+,-./012

C% >/'8) %A)#&) 10! = 3 628 8003 D&+(+E)"1/ () >/'8$() 1% ?&+'(+"5 10! ≈ 3 598 7003

!"#$%&"'(' )"*+! ,-./#(% 0$1(%2(*$!2- '$ 3(2$*42"5(& 67%(& 8 91."5('(& :!$%- ; 3(%??@ 39;AAA> B6%(52"5(C &$*(!( @ B

!"#$%$%&' '()"#%*&' +,#, -, '".,/, 80 1(2#" "- '%)(%"/3" .,3"#%,-4 5/3")#,$%6/ +&# +,#3"'7 ,-)(/,' %/3")#,-"' 3#%)&/&.83#%$,' 9 '('3%3($%6/0 :0 ;".("'3#" -, %*"/3%*,* sec(x) =

sen(x) cos(x) + cos(x) 1 + sen(x)

9 *"'+(2%"'6"4 %'6">.#$"4 ?

R −1

du 1+u2

du −1 1+u2 4

+

π 4

¢

= 2 (arctan(u))1−1 = π2.

*+,-./012 @6%$%&#'() %'6">.#*%)' 5). 5#.6"4 A *)'4%(".#'() *)+) f (x) = ln(x) A g′(x) = dx/ 6"'"+)4 B2" f ′ (x) = dx/x A g(x) = x: #4%/ Z Z xdx = x ln(x) − x + C. ln(x)dx = x ln(x) − x

!?

*+,-./012

@6%$%&#'() %'6">.#*%)' 5). 5#.6"4 A *)'4%(".#'() *)+) f (x) = ln2 (x) A g ′ (x) = dx/ 6"'"+)4 B2" f ′ (x) = 2 ln(x)/x A g(x) = x: #4%/ Z

"?

R

2

2

ln (x)dx = x ln (x) − 2

√tan(x)dx 7 2 sec (x)−4

*+,-./012

Z

x ln(x)dx = x ln2 (x) − 2(x ln(x) − x) + C. x

!"#$ % &'"%&'"()'*

!"###$

tan(x)dx p = sec2 (x) − 4

Z

sen(x)dx q cos2 (x) cos(x) 1−4 cos2 (x) Z sen(x)dx p = 1 − 4 cos2 (x)

Z

!"#$%&#'() "$ *#+,%) (" -#.%#,$" u = cos(x)/ du = − sen(x)dx0 1"'"+)2 34" Z

tan(x)dx p =− sec2 (x) − 4

Z

√

du . 1 − 4u2

!"#$%&#'() "$ *#+,%) (" -#.%#,$" u = cos(θ)/2/ du = − sen(θ)/2dθ0 ),1"'"+)2 34" −

Z

("-)$-%"'() $)2 *#+,%)2

arctan(x)dx6

R

Z

−1 − sen(θ)dθ p θ + C, = 2 2 2 sen (x)

tan(x)dx −1 p arc cos(2 cos(x)) + C. = 2 sec2 (x) − 4

Z

5

du √ = 1 − 4u2

*+,-./012 71%$%&#'() %'1"8.#*%)' 9). 9#.1"2: 2"# f (x) ′

f (x) =

dx dx 1+x2

/ g(x) = x ; Z

arctan(x)dx = x arctan(x) −

Z

= arctan(x) ; g ′ (x) = dx0 #2%/

xdx . 1 + x2

!"#$%&#'() "$ *#+,%) (" -#.%#,$": u = 1 + x2 ; du = 2xdx6 */ Z

2 x

2 x

x e dx = x e − 2

Z

xex dx.

345*6,$4)( 9(, 9$,5*1 (5,$ +*;/ 1*$ f (x) = x/ g ′ (x) = ex dx/ f ′ (x) = dx : g(x) = ex Z

2 x

2 x

µ

x

x e dx = x e − 2 xe −

Z

x

e dx

¶

= x2 ex − 2xex + 2ex .

012/ Z

¡ ¢ (x3 − 2x) exp(x)dx = (x3 − 2x) exp(x) + 2ex − 3 x2 ex − 2xex + 2ex .

!"#$ % &'"%&'"()'*

!"###$R !

e2x −e−2x dx e2x +e−2x

*+,-./012

" Z

!

!

e2x − e−2x dx = e2x + e−2x

Z

1 senh(2x) dx = ln(cosh(2x)) + C. cosh(2x) 2

" *+,-./012 #$%&'(%)*+ $& ,%-.'+ *$ /%0'%.&$ u = e3x1 du = 3e3xdx2 3$ 4'$)$ 56$

R

3x

√e dx 4−e6x

Z du 1 e3x √ √ dx = 6x 3 4−e 4 − u2 1 1 = arc sen(u/2) + C = arc sen(e3x /2) + C. 3 3 R R 7" 8$%) A = exp(sx) cos(tx)dx 9 B = exp(sx) sen(tx)dx" :$-6$340$ 56$ sB + tA = exp(sx) sen(tx) + C ;36 x + 2 = tan(t)1 Z

!<

R√

−x2 + x + 1dx( )67+#+5.*&=

*+,-./012 Z √

"<

R√

−x2

5 + x + 1dx = 8

*+,-./012

R

µ

5 4

− (x − 12 )2 = −x2 + x + 1 > x −

arc sen

µ

2x − 1 √ 5

¶

−

µ

2x − 1 √ 5

1 2

2x−1 dx x2 −6x+13

*+,-./012

Z

R

√

5 2

sen(t)1

¶ √ ¶ 2 1 + x − x2 √ + C. 5

√ ¡ ¢ √ 2 − 6t 2 − 6t 9 ln t − 3 + t t (2t − 6) − + C. t2 − 6tdt = 4 2

( )67+#+5.*&=

2x−1 (x−3)2 +4

=

(2x−6)+5 (x−3)2 +4

1

¢ 5 ¡ 2 2x − 1 − 6x + 13 + arctan dx = ln x x2 − 6x + 13 2

µ

x−3 2

?1 @&%%+ %&) )*76*+5$+) *5$+7#&%+) <

=

t2 − 6tdt( )67+#+5.*&= t − 3 = 3 sec(x)1 Z √

#<

¯ ¯ ¯ ¯ dx 1 ¯ √ dx = ln ¯ √ + (x + 2)¯¯ + C. x2 + 4x + 5 x2 + 4x + 5

sen(4y) cos(5y)dy 1

*+,-./012

Z

sen(4y) cos(5y)dy =

1 −1 cos (9y) + cos(y) + C. 18 2

¶

+ C.

!"#$ % &'"%&'"()'*

!"###$R !

!!

"!

#!

$!

sen4 (3t) cos4 (3t)dt"

*+,-./012

R

Z

−1 1 sen3 (3t) cos5 (3t) − sen (3t) cos5 (3t) 24 48 3x 1 1 sen (3t) cos3 (3t) + sen (3t) cos (3t) + + C. + 192 128 128

sen4 (3t) cos4 (3t)dt =

tan4 (x)dx"

*+,-./012 R

Z

tan4 (x)dx =

1 tan3 (x) − tan(x) + x + C. 3

Z

tan4 (x)dx =

−1 csc2 (x) + ln(tan(x)) + C. 2

tan−3 (x) sec4 (x)dx"

*+,-./012 R

csc3 (y)dy "

*+,-./012

Z

R π/2 π/4

csc3 (y)dy =

−1 1 csc(y) + ln |csc(y) − cot(y)| + C. 2 2

p sen3 (z) cos(z)dz "

*+,-./012 #$%&'(%)*+ $& ,%-.'+ *$ /%0'%.&$ u = cos(z) du = − sen(z)dz Z

π/2

π/4

Z p 3 sen (z) cos(z)dz =

&!

R

3 sen(z) dz cos2 (z)+cos(z)−2

*+,-./012 Z

µ

2 3/2 2 7/2 u − u 3 7

√ (1 − u2 ) udu

¶√2/2

= 0, 3115.

0

" 1 3 sen(z) dz = (ln |cos(z) − 1| − ln |cos(z) + 2|) + C. + cos(z) − 2 3

cos2 (z)

1 234$0$),'%5 1 + cos2 (t) = 2 cos2 (x) + sen2 (x)1 sec2 (t) 1 2 cos (x) + sen2 (x) = cos2 (t)(2 + tan2 (t)) 6 1+cos 2 (t) = 2+tan2 (t) "

R

dt 1+cos2 (t) 2

*+,-./012

Z

'!

2/2

0

=

%!

√

R

√

2 dt = arctan 2 1 + cos (t) 2

Ã√

!

2 tan(t) 2

+ C.

x sen3 (x) cos(x)dx1 234$0$),'%5 37'&',$ ')7$40%,'8) 9+0 9%07$2"

*+,-./012 Z

x sen3 (x) cos(x)dx =

x 1 3 3x sen4 (x) + sen3 (x) cos(x) + sen(x) cos(x) − + C. 4 16 32 32