Example 6 (i) - Chapter 7 Class 12 Integrals (Term 2)
Last updated at Aug. 20, 2021 by Teachoo
Examples
Example 1 (i)
Example 1 (ii)
Example 1 (iii)
Example 2 (i)
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Example 2 (iii) Important
Example 3 (i)
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Example 3 (iii)
Example 4
Example 5 (i)
Example 5 (ii)
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Example 6 (i) You are here
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Example 7 (i)
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Example 7 (iii)
Example 8 (i)
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Example 9 (i)
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Example 10 (i)
Example 10 (ii) Important
Example 11
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Example 16 Important
Example 17
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Example 19
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Example 21 Important
Example 22 Important
Example 23
Example 24
Example 25 Important Deleted for CBSE Board 2023 Exams
Example 26 Deleted for CBSE Board 2023 Exams
Example 27 (i)
Example 27 (ii) Important
Example 27 (iii)
Example 27 (iv) Important
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Example 37
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Example 25 (Supplementary NCERT) Important Deleted for CBSE Board 2023 Exams
Chapter 7 Class 12 Integrals
Serial order wise
Transcript
Example 6 Find the following integrals: (i) ∫1▒〖sin^3𝑥 cos^2𝑥 〗 𝑑𝑥 ∫1▒〖sin^3𝑥 cos^2𝑥 〗 𝑑𝑥 Let cos 𝑥=𝑡 Differentiating both sides 𝑤.𝑟.𝑡.𝑥. −sin𝑥=𝑑𝑡/𝑑𝑥 𝑑𝑥=(−𝑑𝑡)/sin𝑥 Now are equation becomes ∫1▒〖sin^3𝑥 cos^2𝑥 〗 𝑑𝑥 Putting value of 𝑐𝑜𝑠𝑥 and 𝑑𝑥 = ∫1▒sin^3𝑥 .𝑡^2. 𝑑𝑥 = ∫1▒sin^3𝑥 .𝑡^2. 𝑑𝑡/(−sin𝑥 ) = ∫1▒sin^3𝑥/(−sin𝑥 ) 𝑡^2. 𝑑𝑡 = –∫1▒sin^2𝑥 𝑡^2. 𝑑𝑡 = – ∫1▒(1−cos^2𝑥 ) 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^2𝑥=1−cos^2𝑥) = (〖−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〗^𝟑𝒙 +𝟏/𝟓 〖𝒄𝒐𝒔〗^𝟓𝒙 +𝑪
