Mm 202 Integracion Por Partes
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Transcript of Mm 202 Integracion Por Partes
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Guia de Ejercicios,
MM-202
Carlos Cruz
September 10, 2015
Nombre: Registro Estudiantil:
Instrucciones: Resuelva cada ejercicios de forma clara honesta y ordenada mostrando todo su proced-imiento de lo contrario no tendra opcion a creditos1
INTEGRACION POR PARTES
1.
sin(ln x)dx
(a) 12x cos(ln x) sin(ln x) + C (b) 1
2x cos(ln x) 1
2x sin(ln x) + C
(c) 12x cos(ln x) +
1
2x sin(lnx) + C (d) 1
2cos(ln x) +
1
2x sin(ln x) + C
(e) 12cos(ln x) +
1
2sin(lnx) + C
2.
cos(ln x)dx
(a) 12x cos(ln x) sin(ln x) + C (b) 1
2x cos(ln x) 1
2x sin(ln x) + C
(c) 12x cos(ln x) +
1
2x sin(lnx) + C (d) 1
2cos(ln x) +
1
2x sin(ln x) + C
(e) 12cos(ln x) +
1
2sin(lnx) + C
3.
x21 x2dx =
(a)1
2x1 x2 + 1
2sin1 x+ C (b) 1
2x1 x2 1
2sin1 x+ C
(c) 2x1 x2 + 2 sin1 x+ C (d) 1
2x1 x2 + 1
2sin1 x+ C
(e) x1 x2 + 1
2sin1 x+ C
1Disponible Gratuitamente en www.mathunah.wordpress.com
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4.
a2 + x2dx, a R
(a)1
2xa2 + x2 +
a2
4ln(
a2 + x2 + x)+ C (b)
1
2xa2 + x2 + a2 ln
(a2 + x2 + x
)+ C
(c) xa2 + x2 + a2 ln
(a2 + x2 + x
)+ C (d) x
a2 + x2 +
a2
2ln(
a2 + x2 + x)+ C
(e) 2xa2 + x2 +
a2
2ln(
a2 + x2 + x)+ C
5.
x2ex
(x+ 2)2dx
(a)ex(x 2)x+ 2
+ C (b)ex(x+ 2)
x 2 + C (c)ex(x 2)(x+ 2)2
+ C (d)2ex(x 2)x+ 2
+ C
(e)(x 2)ex(x+ 2)
+ C
6.
(sin1 x
)2dx
(a) 21 x2 sin1 x 2x+ x2 sin1 x+ C (b) 2
1 x2 sin1 x 1
2x+ x(sin1 x)2 + C
(c)1 x2 sin1 x 2x+ x(sin1 x)2 + C (d) 2
1 x2 sin1 x 2x+ x(sin1 x)2 + C
(e) 21 x2 sin x 2x+ x(sin1 x)2 + C
7.
xn ln xdx, n 6= 1
(a)xn+1 ln x
n+ 1 x
n+2
(n + 1)(n+ 2)+ C (b)
xn+1 ln x
n + 1 x
n+2
(n+ 1)2+ C
(c)xn+1 ln x
n + 1+
xn+1
(n + 1)2+ C (d)
xn+1 ln x
n + 1 x
n+1
(n+ 1)2+ C
(e)xn ln x
n + 1 x
n+2
(n+ 1)(n+ 2)+ C
8.
tan1(
x)dx
(a) (x+ 1) tan(x)x+ C (b) (x+ 1) tan1(x)x+ C
(c) (x 1) tan1(x)x+ C (d) (x+ 1) tan1(x) +x+ C
(e) x tan1(x)x+ C
9.
sin1 x1 xdx
(a)1 x sin1 x+
1 + x+ C (b)
1 x sin1 x+ 4
1 + x+ C
(c) 21 x sin1 x+1 + x+ C (d) 21 x sin1 x+ 41 + x+ C
(e) 21 x sin1 x+ 41 + x+ C
2
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10.
x tan1 x
1 + x2dx
(a)1 + x tan1 x ln(x+
1 + x2) + C (b)
1 + x2 tan1 x+ ln(x+
1 + x2) + C
(c)1 + x2 tan1 x ln(x+
1 + x2) + C (d)
1 + x2 tan1 x x ln(x+
1 + x2) + C
(e)1 + x2 tan1 x 1
2ln(x+
1 + x2) + C
11.
sin1(
x)
1 x dx
(a) 2x 21 x sin1(x) + C (b) 2x1 x sin1(x) + C
(c) 2x 1
2
1 x sin1(x) + C (d) 2x 2
1 x sin(x) + C
(e)x 21 x sin1(x) + C
12.
exdx
(a) 2ex(x 1) + C (b) 1
2ex(x 1) + C
(c) ex(x 1) + C (d) 2e
x(x 1) + C
(e) 2ex(x 2) + C
13.
sin( 3
x)dx
(a) 3 3x sin( 3
x) 3( 3
x2 2) cos( 3x) + C (b) 6 3x sin( 3x) 3( 3
x2 2) cos( 3x) + C
(c) 6 3x sin( 3
x) ( 3
x2 2) cos( 3x) + C (d) 6 3x sin(x) 3( 3
x2 2) cos(x) + C
(e) 6 3x sin( 3
x) 3( 3
x2 3) cos( 3x) + C
14.
sin1 x(1 x2)3dx
(a)x sin1 x1 x2 + ln(1 x
2) + C (b)sin1 x1 x2 + ln(1 x
2) + C
(c)x sin1 x1 x2 ln(1 x
2) + C (d) 2x sin1 x1 x2 + ln(1 x
2) + C
(e)x sin1 x1 x2 + 2 ln(1 x
2) + C
15.
x2 tan1 x
1 + x2dx
(a) 12ln(x2 + 1) 1
2(tan1 x)2 + x tan1 x+ C (b)
1
2ln(x2 + 1) 1
2(tan1 x)2 + x tan1 x+ C
(c) 14ln(x2 + 1) 1
2(tan1 x)2 + x tan1 x+ C (d) 1
2ln(x2 + 1) 1
2(tan1 x)2 + x tan1 2x+ C
(e) ln(x2 + 1) 12(tan1 x)2 + x tan1 x+ C
3
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16.
tan1 x
x2(1 + x2)dx
(a) 12ln(1 + x2) + ln x 1
2(tan1 x)2 tan
1 x
x+ C
(b) 12ln(1 + x2) + lnx 1
2(tan1 x)2 tan1 x+ C
(c)1
2ln(1 + x2) + ln x 1
2(tan1 x)2 tan
1 x
x+ C
(d) 12ln(1 + x2) + lnx+
1
2(tan1 x)2 tan
1 x
x+ C
(e)1
2ln(1 + x2) ln x+ 1
2(tan1 x)2 +
tan1 x
x+ C
17.
(x2 x+ 1)ex
(x2 + 1)3
2
dx
(a)ex
(x2 + 1)3
2
+ C (b)ex
(x2 + 1)2+ C (c)
ex
x2 + 1+ C (d)
exx2 + 1
+ C (e)xexx2 + 1
+ C
18.
x3ex
2
(x2 + 1)2dx
(a) ex2
x2 + 1+ C (b) 1
2
ex2
x2 + 1+ C (c)
ex2
x2 + 1+ C (d)
1
2
ex2
x2 + 1+ C (e)
1
2
ex
x2 + 1+ C
19.
ex(x 1)
x2dx
(a)ex
x(b) e
x
x+ C (c)
ex
x+ C (d)
ex
x2+ C (e)
ex
x+ C
20.
ln(x+
x2 a2
)dx, a R
(a) x ln(x+
x2 a2
)x2 + a2 + C (b) x ln
(x+
x2 a2
)+x2 a2 + C
(c) x+ ln(x+
x2 a2
)x2 a2 + C (d) x ln
(x+
x2 a2
)x2 a2 + C
(e) x ln(x+
x2 a2
)x a+ C
EJERCICIOS DIVERSOS: Utilizando integracion por partes resuelva las siguientes integrales:
1.
sin1(
x)
xdx 2.
x sin1(3x2)dx 3.
x
+ xdx, , R
4.
xex
(1 + x)2dx 5.
cos1
(x
x+ 1
)dx
6.
ln(x+
a2 + x2
)dx 7.
ln x 1(ln x)2
dx 8.
x ln x(x2 1)3dx
9.
xex1 + ex
dx 10.
x tan1 x
(1 + x2)2dx
4
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FORMULAS DE REDUCCION USANDO INTEGRACION POR PARTES: Usando inte-gracion por partes demuestre la validez de las siguientes integrales
1.
xn cos xdx = xn sin x n
xn1 sin x
2.
1
(a2 + x2)mdx =
1
a2
(x
(2m 2)(a2 + x2)m1 +2m 32m 2
1
(a2 + x2)m1dx
), m 6= 1
3.
sinm xdx = sin
m1 x cosx
m+m 1m
sinm2 xdx
4.
sinm x cosn xdx =
sinm+1 x cosn1 x
m+ n+
n 1m+ n
sinm x cosn2 xdx, m 6= n
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