Check sibling questions

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


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

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Davneet Singh

Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 12 years. He provides courses for Maths, Science, Social Science, Physics, Chemistry, Computer Science at Teachoo.