Example 27 - Chapter 7 Class 12 Integrals - Part 11

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Example 27 - Chapter 7 Class 12 Integrals - Part 12

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Example 27 - Chapter 7 Class 12 Integrals - Part 13

  1. Chapter 7 Class 12 Integrals (Term 2)
  2. Serial order wise

Transcript

Example 27 Evaluate the following integrals: (iv) ∫_0^(πœ‹/4)β–’γ€–sin^3⁑2𝑑 cos⁑2 𝑑〗 𝑑𝑑 Let F(π‘₯)=∫1▒〖𝑠𝑖𝑛^3 2𝑑 π‘π‘œπ‘  2𝑑 𝑑𝑑〗 Let s𝑖𝑛 2𝑑=𝑒 Differentiating w.r.t.π‘₯ (𝑑(sin⁑2𝑑))/𝑑𝑑=𝑑𝑒/𝑑𝑑 2cπ‘œπ‘  2𝑑 =𝑑𝑒/𝑑𝑑 𝑑𝑑=𝑑𝑒/(2 π‘π‘œπ‘  2𝑑) Putting value of u and du in our integral ∫1▒〖𝑠𝑖𝑛^3 2𝑑 π‘π‘œπ‘  2𝑑 𝑑𝑑〗=∫1▒〖𝑒^3 π‘π‘œπ‘  2𝑑 Γ— 𝑑𝑒/(2 π‘π‘œπ‘  2𝑑)γ€— =1/2 ∫1▒〖𝑒^3 𝑑𝑒〗 =1/2 𝑒^(3+1)/(3+1)=1/2 𝑒^4/4= 𝑒^4/8 Putting back 𝑒=𝑠𝑖𝑛 2𝑑 =1/8 𝑠𝑖𝑛^4 2𝑑 Hence, F(𝑑)=1/8 𝑠𝑖𝑛^4 2𝑑 Now, ∫_0^(πœ‹/4)▒〖𝑠𝑖𝑛^3 2𝑑 π‘π‘œπ‘  2𝑑=𝐹(πœ‹/4)βˆ’πΉ(0) γ€— =1/8 𝑠𝑖𝑛^4 2(πœ‹/4)βˆ’1/8 𝑠𝑖𝑛^4 2(0) =1/8 𝑠𝑖𝑛^4 πœ‹/2βˆ’1/8 𝑠𝑖𝑛^4 (0) =1/8 Γ—1^4βˆ’1/8 Γ—0^4 =1/8 Γ—1βˆ’0 =𝟏/πŸ–

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Davneet Singh
Davneet Singh is a graduate from Indian Institute of Technology, Kanpur. He has been teaching from the past 10 years. He provides courses for Maths and Science at Teachoo.