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Example 30 - Evaluate integral -1 -> 2 |x3 - x| dx - Definate Integration by properties - P2

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Example 30 Evaluate ﷐−1﷮2﷮﷐﷐𝑥﷮3﷯−𝑥﷯﷯ 𝑑𝑥 ﷐﷐𝑥﷮3﷯−𝑥﷯=﷐𝑥﷐﷐𝑥﷮2﷯−1﷯﷯ =﷐𝑥﷯ ﷐﷐𝑥﷮2﷯−1﷯ =﷐𝑥﷯ ﷐﷐𝑥−1﷯﷐𝑥+1﷯﷯ =﷐𝑥﷯ ﷐𝑥−1﷯﷐𝑥+1﷯ Thus, 𝑥=0, 𝑥=1 & 𝑥=−1 ∴ ﷐﷐𝑥﷮3﷯−𝑥﷯= ﷐﷐﷐−𝑥﷯﷐−(𝑥−1)﷯﷐−(𝑥+1)﷯ 𝑖𝑓 𝑥<−1﷮﷐−𝑥﷯(−(𝑥−1))﷐𝑥+1﷯ 𝑖𝑓 −1≤𝑥<0﷮﷐𝑥﷯(−(𝑥−1))﷐𝑥+1﷯ 𝑖𝑓 0≤𝑥<1﷮﷐𝑥﷯﷐𝑥−1﷯﷐𝑥+1﷯ 𝑖𝑓 𝑥≥1﷯﷯ ∴ ﷐﷐𝑥﷮3﷯−𝑥﷯= ﷐﷐−(﷐𝑥﷮3﷯−𝑥) 𝑖𝑓 𝑥<−1﷮(﷐𝑥﷮3﷯−𝑥) 𝑖𝑓 −1≤𝑥<0﷮−﷐﷐𝑥﷮3﷯−𝑥﷯ 𝑖𝑓 0≤𝑥<1﷮﷐﷐𝑥﷮3﷯−𝑥﷯ 𝑖𝑓 𝑥≥1﷯﷯ ﷐−1﷮2﷮﷐﷐𝑥﷮3﷯−𝑥﷯﷯𝑑𝑥=﷐−1﷮0﷮﷐﷐𝑥﷮3﷯−𝑥﷯﷯𝑑𝑥−﷐0﷮1﷮﷐﷐𝑥﷮3﷯−𝑥﷯﷯𝑑𝑥+﷐1﷮2﷮﷐﷐𝑥﷮3﷯−𝑥﷯﷯𝑑𝑥 =﷐−1﷮0﷮﷐𝑥﷮3﷯﷯𝑑𝑥−﷐−1﷮0﷮𝑥﷯𝑑𝑥−﷐0﷮1﷮﷐𝑥﷮3﷯﷯𝑑𝑥+﷐0﷮1﷮𝑥﷯𝑑𝑥+﷐1﷮2﷮﷐𝑥﷮3﷯﷯𝑑𝑥−﷐1﷮2﷮𝑥﷯𝑑𝑥 =﷐﷐﷐﷐𝑥﷮4﷯﷮4﷯﷯﷮−1﷮0﷯−﷐﷐﷐﷐𝑥﷮2﷯﷮2﷯﷯﷮−1﷮0﷯−﷐﷐﷐﷐𝑥﷮4﷯﷮4﷯﷯﷮0﷮1﷯+﷐﷐﷐﷐𝑥﷮2﷯﷮2﷯﷯﷮0﷮1﷯+﷐﷐﷐﷐𝑥﷮4﷯﷮4﷯﷯﷮1﷮2﷯−﷐﷐﷐﷐𝑥﷮2﷯﷮2﷯﷯﷮1﷮2﷯ =﷐﷐0 − ﷐﷐−1﷯﷮4﷯﷮4﷯﷯−﷐﷐0 − ﷐﷐−1﷯﷮2﷯﷮2﷯﷯−﷐﷐﷐﷐1﷯﷮4﷯ − 0﷮4﷯﷯+﷐﷐﷐﷐1﷯﷮2﷯ − 0﷮2﷯﷯+﷐﷐﷐﷐2﷯﷮4﷯ − ﷐﷐1﷯﷮4﷯﷮4﷯﷯−﷐﷐﷐﷐2﷯﷮2﷯ − ﷐﷐1﷯﷮2﷯﷮2﷯﷯ =﷐−1﷮4﷯ − ﷐﷐−1﷮2﷯﷯ − ﷐1﷮4﷯ + ﷐1﷮2﷯ + ﷐﷐16 − 1﷮4﷯﷯−﷐﷐4 − 1﷮2﷯﷯ =﷐−1﷮4﷯ + ﷐1﷮2﷯ − ﷐1﷮4﷯ + ﷐1﷮2﷯ +﷐15﷮4﷯ − ﷐3﷮2﷯ =﷐−1 + 2 − 1 + 2 + 15 − 6﷮4﷯ =﷐𝟏𝟏﷮𝟒﷯

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