Example 6 - Find integrals (i) sin^3 x cos^2 x dx (ii) sin x

Example 6 - Chapter 7 Class 12 Integrals - Part 2
Example 6 - Chapter 7 Class 12 Integrals - Part 3

Learn in your speed, with individual attention - Teachoo Maths 1-on-1 Class


Transcript

Example 6 Find the following integrals: (i) ∫1▒〖sin^3⁡𝑥 cos^2⁡𝑥 〗 𝑑𝑥 ∫1▒〖sin^3⁡𝑥 cos^2⁡𝑥 〗 𝑑𝑥 Let cos 𝑥=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥. −sin⁡𝑥=𝑑𝑡/𝑑𝑥 𝑑𝑥=(−𝑑𝑡)/sin⁡𝑥 Now are equation becomes ∫1▒〖sin^3⁡𝑥 cos^2⁡𝑥 〗 𝑑𝑥 Putting value of 𝑐𝑜𝑠⁡𝑥 and 𝑑𝑥 = ∫1▒sin^3⁡𝑥 .𝑡^2. 𝑑𝑥 = ∫1▒sin^3⁡𝑥 .𝑡^2. 𝑑𝑡/(−sin⁡𝑥 ) = ∫1▒sin^3⁡𝑥/(−sin⁡𝑥 ) 𝑡^2. 𝑑𝑡 = –∫1▒sin^2⁡𝑥 𝑡^2. 𝑑𝑡 = – ∫1▒(1−cos^2⁡𝑥 ) 𝑡^2. 𝑑𝑡 = – ∫1▒(1−𝑡^2 ) 𝑡^2. 𝑑𝑡 = – ∫1▒(𝑡^2−𝑡^4 ) 𝑑𝑡 = ∫1▒(−𝑡^2+𝑡^4 ) 𝑑𝑡 = ∫1▒〖−𝑡^2 〗. 𝑑𝑡 + ∫1▒𝑡^4 . 𝑑𝑡 (∴ sin^2⁡𝑥=1−cos^2⁡𝑥) = (〖−𝑡〗^2+1)/(2 + 1)+𝑡^(4 + 1)/(4 + 1)+𝐶 = (−𝑡^3)/3 +𝑡^5/5 +𝐶 Putting back value of t = cos x = (−𝟏)/𝟑 〖𝒄𝒐𝒔〗^𝟑⁡𝒙 +𝟏/𝟓 〖𝒄𝒐𝒔〗^𝟓⁡𝒙 +𝑪

Ask a doubt
Davneet Singh's photo - Co-founder, Teachoo

Made by

Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 13 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.