Answered: a) fe* cos 2x dc (parts)

Advanced Engineering Mathematics

10th Edition

ISBN:9780470458365

Author:Erwin Kreyszig

Publisher:Erwin Kreyszig

Chapter2: Second-order Linear Odes

Section: Chapter Questions

Problem 1RQ

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Can someone help me Integrate these problems? There are specific techniques that have to be used. The first one is integration by parts. The second is integration by partial fractions. I'm not sure for the third one and the fourth is integration by trigonometric substitution. I've attached the first.

Expert Solution

Step 1

$\int e^{x} \cos (2 x) d x S o l v i n g i n t e g r a t i o n b y p a r t s w e g e t, \int u d v = u v - \int v d u L e t u = e^{x} a n d d v = \cos (2 x) = e^{x} \frac{1}{2} \sin (2 x) - \int e^{x} \frac{1}{2} \sin (2 x) d x = e^{x} \frac{1}{2} \sin (2 x) - \frac{1}{2} \int e^{x} \sin (2 x) d x . . . . . . . . . (t a k i n g c o n s \tan t o u t)$

Step 2

$A g a i n a p p l y i n g i n t e g r a t i o n b y p a r t s w e g e t, L e t u = e^{x} a n d d v = \sin (2 x) = e^{x} \frac{1}{2} \sin (2 x) - \frac{1}{2} (- \frac{1}{2} e^{x} \cos (2 x) - \int - \frac{1}{2} e^{x} \cos (2 x) d x) = e^{x} \frac{1}{2} \sin (2 x) - \frac{1}{2} (- \frac{1}{2} e^{x} \cos (2 x) - (- \frac{1}{2} \int e^{x} \cos (2 x) d x))$

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