Example 31 - Chapter 7 Class 12 Integrals

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﷯