Example 43 - Evaluate integral |x sin (pi x)| dx - Examples

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Example 43 (Introduction) Evaluate −1﷮ 3﷮2﷯﷮ 𝑥 sin﷮ 𝜋 𝑥﷯﷯﷯﷯𝑑𝑥 We have 𝑥 sin﷮𝜋𝑥 ﷯, where −1< 𝑥 < 3﷮2﷯ (i) 𝒙 𝑥 < 0 −1≤ 𝑥 < 0 𝑥 > 0 0 ≤ 𝑥 < 3﷮2﷯ (ii) 𝐬𝐢𝐧﷮𝝅 𝒙﷯ −1≤𝑥≤ 3﷮2﷯ −𝜋 ≤𝜋 𝑥≤ 3𝜋﷮2﷯ Let 𝜃=𝜋𝑥 ∴ −𝜋 ≤θ≤ 3𝜋﷮2﷯ From graph it is seen, sin﷮𝜃<0﷯ −𝜋 ≤ 𝜃<0 sin﷮𝜃>0﷯ 0≤ 𝜃<𝜋 sin﷮𝜃<0﷯ −𝜋 ≤ 𝜃 ≤ 3𝜋﷮2﷯ That is, Hence, we have Example 43 Evaluate −1﷮ 3﷮2﷯﷮ 𝑥 sin﷮ 𝜋 𝑥﷯﷯﷯﷯𝑑𝑥 Hence, 𝑥 𝑠𝑖𝑛 (π𝑥)﷯= −𝑥 sin﷮𝜋𝑥﷯ −1 ≤ 𝑥 ≤ 1﷮&−𝑥 sin﷮𝜋𝑥﷯ 1 ≤ 𝑥 ≤ 3﷮2﷯﷯﷯ ∴ −1﷮ 3﷮2﷯﷮ 𝑥 sin﷮ 𝜋 𝑥﷯﷯﷯﷯𝑑𝑥 = −1﷮1﷮𝑥 sin﷮ 𝜋𝑥 𝑑𝑥 − 1﷮ 3﷮2﷯﷮𝑥 sin﷮ (﷯﷯﷯﷯𝜋𝑥) 𝑑𝑥 Solving ﷮﷮𝒙 𝒔𝒊𝒏﷮ 𝝅𝒙 𝒅𝒙﷯﷯ separately ﷮﷮𝑥 sin﷮ 𝜋𝑥 𝑑𝑥﷯﷯ ﷮﷮𝑥 sin﷮𝜋𝑥 𝑑𝑥= ﷮﷮(𝑥)( sin﷮𝜋 𝑥) 𝑑𝑥﷯﷯﷯﷯ = x ﷮﷮ sin﷮𝜋𝑥− ﷮﷮ 𝑑 𝑥﷯﷮𝑑𝑥﷯ ﷮﷮ sin﷮𝜋𝑥 ﷯﷯﷯﷯﷯﷯ 𝑑𝑥 = x (− cos﷮𝜋𝑥)﷯﷮𝜋﷯− ﷮﷮1 − cos﷮𝜋𝑥﷯﷮𝜋﷯﷯﷯ 𝑑𝑥 = − 𝑥 cos﷮𝜋𝑥 ﷯﷮𝜋﷯+ ﷮﷮ cos﷮𝜋𝑥 ﷯﷮𝜋﷯﷯ 𝑑𝑥 = − 𝑥 cos﷮𝜋𝑥 ﷯﷮𝜋﷯+ sin﷮𝜋𝑥 ﷯﷮ 𝜋﷮2﷯﷯ Putting limits Hence answer = 2﷮𝜋﷯− −1﷮ 𝜋﷮2﷯﷯+ 1﷮𝜋﷯﷯ = 2﷮𝜋﷯+ 1﷮ 𝜋﷮2﷯﷯+ 1﷮𝜋﷯ = 𝟑﷮𝝅﷯+ 𝟏﷮ 𝝅﷮𝟐﷯﷯

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