Logo Studenta

ejercicios-resueltos-de-integral-2-integrales-por-sustitucic3b3n-o-cambio-de-variable

¡Este material tiene más páginas!

Vista previa del material en texto

http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
Integrales por Sustitución (Cambio de Variable) 
 
Sección �1 Funciones algebraicas, trigonométricas y logarítmicas 
 
 
�40 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
� �� + �� �� = 1� � �� + �� � �� = 1� � √� �� = 1� ∙ �
����12 + 1 + � =
1� ∙ �
��32 + � =
23� ��� + � 
 
= 23� ��� + � = 23� �(� + ��)� + � 
 � = � + �� �� = � �� 
 
�41 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
 � √5� − 3 �� = 15 � √5� − 3 5 �� = 15 � √� �� = 15 ∙ �
����12 + 1 + � =
15 ∙ �
��32 + � =
215 ��� + � = 215 ��� + � 
 
= 215 �(5� − 3)� + � 
 
 � = 5� − 3 �� = 5 �� 
 
 
�42 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� � ��√��� + � = 12� � 2�� ��√��� + � = 12� � ��√� = 12� � ��
���� = ��
����
− 12 + 1 + � 
 
= �
��12 + � = 2√� + � = 2
���� + � + � 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
 � = ��� + � �� = 2�� �� 
 
�43 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� ��√9� − 1� = 19 � 9��√9� − 1� = 19 � ��√�� = 19 � ��
���� = 19 ∙ �
�����
− 13 + 1 + � =
19 ∙ �
��23 + � 
 
= 318 ��� + � = 16 ���� + � = 16 �(9� − 1)�� + � 
 � = 9� − 1 �� = 9�� 
 
�44 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
�" √� − 4$� �� = � %� − 8��� + 16& �� = � � �� − 8 � ��� �� + 16 � �� = 
 
= ��2 − 8�
��32 + 16� + � =
��2 − 16√�
�
3 + 16� + � 
 
�45 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� � ��(3�� − 4)' = 16 � 6� ��(3�� − 4)' = 16 � ���' = 16 � ��'�� = 16 ∙ �
�'��
−4 + 1 + � = 
 
= 16 ∙ �
��
−3 + � = − 118�� + � = − 118(3�� − 4)� + � 
 � = 3�� − 4 �� = 6� �� 
 
 
�46 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
� 5 ��(3� − 4)� = 53 �
35 ∙ 5 ��(3� − 4)� = 53 � 3 ��(3� − 4)� = 53 � ���� = 53 ∙ �
����
−2 + 1 + � 
 
= 53 ∙ �
��
−1 + � = − 53� + � = − 53(3� − 4) + � 
 � = 3� − 4 �� = 3 �� 
 
�47 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� 8� ��(2�� + 5)' = 2 � 4� ��(2�� + 5)' = 2 � ��(�)' = 2 � ��'�� = 2�
�'��
−4 + 1 + � 
 
= 2���−3 + � = − 23�� + � = − 23(2�� + 5)� + � 
 � = 2�� + 5 �� = 4� �� 
 
�48 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� "√� − �$�√� �� = 2 � "√� − �$
�
2√� �� = 2 � ���� = 2�
�
3 + � = 2(√� − �)
�
3 + � 
 � = √� − � 
 
�� = 12√� �� 
 
�49 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� ���� + � = 1� � � ���� + � = 1� � ��� = 1� ln � + � = 1� ln|�� + �| + � 
 � = �� + � �� = � �� 
 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
�50 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� � ��3�� − 4 = 16 � 6� ��3�� − 4 = 16 � ��� = 16 ln � + � = 16 ln|3�� − 4| + � 
 � = 3�� − 4 �� = 6� �� 
 
�51 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� ��� + 3 = � ��� = ln � + � = ln|� + 3| + � 
 � = � + 3 �� = �� 
 
 
�52 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� 4� ��2�� − 6 = � ��� = ln � + � = ln|2�� − 6| + � 
 � = 2�� − 6 �� = 4� �� 
 
�53 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� (2� − 3)��(�� − 3� + 6)� = � ���� = �
����
−2 + 1 + � = − 1� + � = − 1�� − 3� + 6 + � 
 � = �� − 3� + 6 �� = (2� − 3)�� 
 
�54 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
�(�� − 2)��� − 6� + 3�� = 13 � 3(�� − 2)��� − 6� + 3�� = 13 � √� �� = 13 �
����12 + 1 + � = 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
 
13 �
��32 + � =
29 ��� + � = 29 �(�� − 6� + 3)� + � 
 � = �� − 6� + 3 �� = (3�� − 6)�� 
 
�55 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� �+����(��+ + �), = 1�� � ���
+����(��+ + �), = 1�� � ���, = 1�� � ��,�� = 1�� ∙ �
��,
1 − � + � = (��
+ + �)��,�� − ��� + � 
 � = ��+ + � �� = ���+���� 
 
�56 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� -�.(1 − -�.)��� = − 13 � −3-�.(1 − -�.)��� = − 13 � ���� = − 13 ∙ �
���
2 + 1 + � 
 
= − 13 ∙ �
�
3 + � = − 19 (1 − -�.)� + � 
 � = 1 − -�. �� = −3-�.�� 
 
�57 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� (4 − /�|� + 3|)� ��� + 3 = � ���� = �
���
3 + 1 + � = �
'
4 + � = (4 − /�|� + 3|)
'
4 + � 
 � = 4 − /�|� + 3| 
 
�� = ��� + 3 
 
�58 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
� cos 4� (1 − 3-� 4�)��� = − 14 � −4 cos 4� (1 − 3-� 4�)��� = − 14 � �� �� = − 14 ∙ �
���
3 + 1 + � 
 
= − 116 �' + � = − 116 (1 − 3-� 4�)' + � 
 � = 1 − 3-� 4� �� = −4453 4� �� 
 
�59 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� csc� �√3 + cot � �� = � √� �� = �
����12 + 1 + � =
���32 + � =
23 ��� + � = 23 �(3 + cot � )�+ � 
 � = 3 + cot � �� = − csc� � �� 
 
�60 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� sec 2� tan 2�√1 − sec 2� �� = − 12 � −2 sec 2� tan 2�√1 − sec 2� �� = − 12 � ��
���� = − 12 ∙ �
�����
− 12 + 1 + � 
 
= − 12 ∙ �
��12 + � = −√� + � = −√1 − sec 2� + � 
 � = 1 − sec 2� �� = − 2sec 2� tan 2� �� 
 
�61 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� cos ��1 − 3-� �� �� = − 1� � −� cos ��1 − 3-� �� �� = − 1� � ��� = − 1� ln|�| + � = − 1� ln|1 − 3-� ��| + � 
 � = 1 − 3-� �� �� = −� 453 �� �� 
 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
�62 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� -√.�-√. − 1√� �� = 2 � -
√.�-√. − 12√� �� = 2 � √� �� = 2�
9����:12 + 1 + � =
2���32 + � =
43 ��� + � 
 
= 43 ;(-√. − 1)� + � 
 � = -√. − 1 
�� = -√. ∙ 12√� �� 
 
�63 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� cot � (2 + /� |3-� �|)�� = � � �� = ��2 + � = ( 2 + /� |3-� �|)
�
2 + � 
 � = 2 + /� |3-� �| �� = cot � �� 
 
�64 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� 3-� 2� ��(1 − cos� �)� = 2 � 3-� � cos � ��(3-���)� = 2 � cos � ��3-�< � = 2 � ��< �� = 2�
�<��
−5 + 1 + � = 2�
�'
−4 + � 
 
= − 12� + � = − 12 3-� � + � = − 12 csc � + � 
 3-��� = 1 − cos � 3-� 2� = 2 3-� � cos � � = 3-� � �� = cos � �� 
 
�65 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� 3-���� cos �� �� = 1� � 3-���� ∙ � cos �� �� = 1� � � �� = �
�
2� + � = 3-�
���2� + � 
 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
� = 3-� �� �� = � cos �� �� 
 
�66 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� cot �� csc� �� �� = − 1� � −� cot �� csc� �� �� = − 1� � � �� = − �
�
2� + � = − 45�
���2� + � 
 � = cot � �� = −� csc� �� �� 
 
�67 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� cos� 4� 3-� 4� �� = − 14 � −4 cos� 4� 3-� 4� �� = − 14 � �� �� = − 14 ∙ �
���
2 + 1 + � = − 112 �� + � 
 
= − 112 453�4� + � 
 
 � = cos 4� 
 �� = −4 3-� 4� �� 
 
�68 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� cos 5�√3-� 5� + 4 �� = 15 � 5 cos 5�√3-� 5� + 4 �� = 15 � ��
�� �� = 15 ∙ �
��12 + � =
25 √� + � = 25 √3-� 5� + 4 + � 
 � = 3-� 5� + 4 
 �� = 5 cos 5� �� 
 
�69 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� 4� + 2� + 2 �� = 4 � � +
12� + 2 �� = 4 � (� + 2) −
32� + 2 �� = 4 � �� − 6 � ��� + 2 = 4� − 6 ln|� + 2| + � 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
 � = � + 2 �� = �� 
 
�70 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� (3�� + 2)��� − 1 = � %3� + 3 + 5� − 1& �� = 3 � � �� + 3 � �� + 5 � ��� − 1 
 
= 3��2 + 3� + 5 ln|� − 1| + � 
 � = � − 1 �� = �� 
 
�71 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� ��� /�� � = � ��� �� = �
����
−2 + 1 + � = − 1� + � = − 1ln � + � 
 
 � = ln � 
�� = 1� �� 
 
�72 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� ��2� /� 3� = 12 � ���(ln 3 + ln �) = 12 � ��� = 12 ln � + � = 12 ln|ln 3�| + � 
 ln 3� = ln 3 + ln � 
 � = ln 3 + ln � 
�� = 1� �� 
 
�73 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
� �+���+�� + � �� = 1�(� + 1) � �(� + 1)�+���+�� + � �� = 1�(� + 1) � ��� �� = 
 
= 1�(� + 1) ∙ �
����12 + 1 + � =
1�(� + 1) ∙ �
��32 + � =
2√��3�(� + 1) + � = 2�(��
+�� + �)�3�(� + 1) + � 
 � = ��+�� + � �� = �(� + 1)�+�� 
 
�74 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� 1�� =1 − 1�� �� = 12 � 2�� =1 − 1�� �� = 12 � ��� �� = 12 ∙ �
����12 + 1 + � =
12 ∙ �
��32 + � 
 
= 13 ��� + � = 13 =%1 − 1��&
� + � 
 
� = 1 − 1�� �� = 2����� 
 
�75 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� csc� 3� cos 3� �� = � cos 3�3-��3� �� = 13 � 3 cos 3�3-��3� �� = 13 � ����� = 13 > �
����
−2 + 1? + � = 13 >�
��
−1 ? + � 
 
= − 13� + � = − 13 3-� 3� + � 
 � = 3-� 3� �� = 3 cos 3� �� 
 
�76 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� % 2(� + 1)� − 3(� + 1)� + 4� + 1& �� = 2 � ��(� + 1)� − 3 � ��(� + 1)� + 4 � ��(� + 1) 
 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
= 2 � ��� �� − 3 � ��� �� + 4 � ��� = 2�
��
−2 − 3�
��
−1 + 4 ln|�| + � 
 
= − 1�� + 3� + 4 ln|�| + � = − 1(� + 1)� + 3� + 1 + 4 ln|� + 1| + � 
 � = � + 1 �� = �� 
 
�77 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� % 3� + 2 − 4� + 5& �� = 3 � ��� + 2 − 4 � ��� + 5 = 3 ln|� + 2| − 4 ln|� + 5| + � 
 
 
�78 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
 
� % 32� − 1 + 53� − 4& �� = 3 � ��2� − 1 + 5 � ��3� − 4 = 32 � 2��2� − 1 + 53 � 3��3� − 4 = 32 ln|2� − 1| + 53 ln|3� − 4| + � 
 � = 2� − 1 �� = 2 �� @ = 3� − 4 �� = 3 �� 
 
�79 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
 
� 3-� �√cos� �� �� = − � − 3-� �√cos� �� �� = − � ��
�� �� = − ������− 23 + 1 + �= −
���13 + � = −3√�
� + � = −3√cos �� + � 
 � = cos � �� = −3-� � �� 
 
�80 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
� 3-��� 3-� 2� �� = � 3-��� ∙ 2 3-� � cos � �� = 2 � 3-�'� ∙ cos � �� = 2 � �' �� = 2�<5 + � = 23-�
<�5 + � 
 3-� 2� = 2 3-� � cos � � = 3-� � �� = cos � �� 
 
�81 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� �A3-��√1 − 45�A = � ��
�� �� = ��
����
− 12 + 1 + � =
���12 + � = 2√� + � = 2√1 − 45�A + � 
 13-��A = csc� A � = 1 − 45�A �� = csc� A �A 
 
 
�82 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
 
� 3 3-� � cos � �1 − 23-�� � = 3 �
3-� � cos � 
�1 − 23-�� � =
32 � 23-� � cos � �1 − 23-�� � =
32 � 3-� 2��cos 2� = −
34 � −2 3-� 2��cos 2� = −
34 � ������ 
 
= − 34 �
�����
− 12 + 1 + � = −
34 �
��12 + � = −
32 √� + � = − 32 �cos 2� + � 
 1 − 23-�� � = cos 2� 2 3-� � cos � = 3-� 2� 
 � = cos 2� �� = −23-� 2� 
 
�83 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
B √1 + cos � �� = B √1 + cos � ∙ √��CDE .√��CDE . �� = B √��CDEF .√��CDE . �� = B √GH+F.√��CDE . �� = B GH+ .√��CDE . �� 
 
http://CursoDeCalculo.Wordpress.com 
Profesor Raúl Vega Muñoz 
 
Este documento forma parte del Curso Gratuito de Cálculo Diferencial e Integral de http://CursoDeCalculo.Wordpress.com inscríbete gratis ahora 
mismo y compártelo con tus amigos de Facebook! 
 
= � ���� �� = ��
����
− 12 + 1 + � =
���12 + � = 2√� + � = 2√1 − cos � + � 
 � = 1 − cos � �� = 3-� � �� 
 
�84 -------------------------------------------------------------------------------------------------------------------------------------------------------------
- 
 
� 3-�
�'�
cos��' � �� = �
3-��'�
cos�' � ∙ cos� � �� = � �tan� �
I ∙ sec� � �� = � ��' �� = �
�'��34 + 1 + � =
�J'74 + � 
 
= 47 ��JI + � = 47 ����J�I + � 
 � = tan � �� = sec� � ��

Continuar navegando