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Example 27 - Chapter 7 Class 12 Integrals - Part 11

Example 27 - Chapter 7 Class 12 Integrals - Part 12
Example 27 - Chapter 7 Class 12 Integrals - Part 13

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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 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.