Example 29 - Chapter 7 Class 12 Integrals

Last updated at Dec. 16, 2024 by Teachoo

Example 29 - Evaluate definite integral sin2 x dx - Examples - Examples

part 2 - Example 29 - Examples - Serial order wise - Chapter 7 Class 12 Integrals

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Example 29 Evaluate ∫_((−𝜋)/4)^(𝜋/4)▒sin^2⁡𝑥 𝑑𝑥 Let f(x) = 〖𝑠𝑖𝑛〗^2 𝑥 f(-x) = 〖𝑠𝑖𝑛〗^2 (−𝑥)=(−sin⁡𝑥 )^2=〖𝑠𝑖𝑛〗^2 𝑥 Since f(x) = f(-x) Hence, 〖𝑠𝑖𝑛〗^2 𝑥 is an even function ∫_((−𝜋)/4)^(𝜋/4)▒sin^2⁡𝑥 𝑑𝑥=∫_0^(𝜋/4)▒sin^2⁡𝑥 𝑑𝑥 = ∫_0^(𝜋/4)▒((1 − cos⁡〖2 〗 𝑥)/2) 𝑑𝑥 = 2∫_0^(𝜋/4)▒〖[1/2−(cos⁡2 𝑥)/2] 𝑑𝑥〗 = 2 [𝑥/2−sin⁡2𝑥/(2×2)]_0^(𝜋/4) = 2 [𝑥/2−sin⁡2𝑥/4]_0^(𝜋/4) Putting Limits = 2(𝜋/4 (1/2)−sin⁡2(𝜋/4)/4) – 2 (0/2−sin⁡2(0)/4) = 2(𝜋/8 −⁡〖sin⁡(𝜋/2)/4〗 )−0 = 2 (𝜋/8−1/4) = 𝝅/𝟒−𝟏/𝟐

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Davneet Singh has done his B.Tech from Indian Institute of Technology, Kanpur. He has been teaching from the past 14 years. He provides courses for Maths, Science and Computer Science at Teachoo

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