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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� � ��
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