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


Example 25 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 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.