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Example 31 - Evaluate definite integral sin2 x dx - Definate Integration by properties - P7

Example 31 - Chapter 7 Class 12 Integrals - Part 2


Transcript

Example 31 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

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.