Example 21
Last updated at Dec. 8, 2016 by Teachoo
Transcript
Example 21 Find 𝑒𝑥 sin𝑥 𝑑𝑥 Let I1 = 𝑒𝑥 sin𝑥 𝑑𝑥 Hence, we take First function :- 𝑓 𝑥= sin𝑥 Second function :- g 𝑥= 𝑒𝑥 I1 = sin𝑥 𝑒𝑥 𝑑𝑥− 𝑑 sin𝑥𝑑𝑥 𝑒𝑥 𝑑𝑥 𝑑𝑥 I1 = 𝑒𝑥 sin𝑥− cos𝑥 . 𝑒𝑥 𝑑𝑥 Let I2 = cos𝑥 . 𝑒𝑥 𝑑𝑥 Hence, we take First function :- 𝑓 𝑥= cos𝑥 Second function :- g 𝑥= 𝑒𝑥 I2 = cos x 𝑒𝑥 𝑑𝑥 – (( cos𝑥)′ 𝑒𝑥𝑑𝑥)𝑑𝑥 I2 = cos x 𝑒𝑥 – (− sin𝑥) 𝑒𝑥 𝑑𝑥 I2 = 𝑒𝑥 cos x + sin𝑥 𝑒𝑥 𝑑𝑥 I2 = 𝑒𝑥 cos x + 𝐼1 Now, Putting value of I2 in (1) , I1 = 𝑒𝑥 sin𝑥− cos𝑥 𝑒𝑥 𝑑𝑥 I1 = 𝑒𝑥 sin𝑥− 𝑒𝑥 cos𝑥+𝐼1 I1 = 𝑒𝑥 sin𝑥− 𝑒𝑥 cos𝑥−𝐼1 2I1 = 𝑒𝑥 sin𝑥− 𝑒𝑥 cos𝑥 I1 = 12 𝑒𝑥 sin𝑥− 𝑒𝑥 cos𝑥 + C I1 = 𝑒𝑥2 sin𝑥− cos𝑥 + C I1 = 𝑒𝑥 sin𝑥− 𝑒𝑥 cos𝑥+ 𝑒𝑥 sin𝑥 𝑑𝑥+𝐶1 I1 = 𝑒𝑥 sin𝑥− 𝑒𝑥 cos𝑥+ 𝑒𝑥 sin𝑥 𝑑𝑥−𝐶1 I1 = 𝑒𝑥 sin𝑥− 𝑒𝑥 cos𝑥−𝐼1−𝐶1 2𝐼1 = 𝑒𝑥 sin𝑥− 𝑒𝑥 cos𝑥−𝐶1 I1 = 𝑒𝑥 sin𝑥 − 𝑒𝑥 cos𝑥2 + 𝐶1 2 I1 = 𝒆𝒙𝟐( 𝒔𝒊𝒏𝒙 − 𝒄𝒐𝒔𝒙)+𝐂
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About the Author
Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He provides courses for Mathematics from Class 9 to 12. You can ask questions here.