Misc 29 - Definite integral dx / root 1+x - root x - Miscellaneous

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  1. Chapter 7 Class 12 Integrals
  2. Serial order wise
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Misc 29 Evaluate the definite integral ﷐0﷮1﷮﷐𝑑𝑥﷮﷐﷮1 + 𝑥﷯ − ﷐﷮𝑥﷯﷯ ﷯ ﷐0﷮1﷮﷐𝑑𝑥﷮﷐﷮1 + 𝑥﷯ − ﷐﷮𝑥﷯﷯ ﷯ Rationalizing i.e., multiplying and dividing by ﷐﷐﷮1 + 𝑥﷯ −﷐﷮𝑥﷯﷯ = ﷐0﷮1﷮﷐𝑑𝑥﷮﷐﷮1 + 𝑥﷯ − ﷐﷮𝑥﷯﷯ ﷯×﷐﷐﷮1 + 𝑥﷯ + ﷐﷮𝑥﷯﷮﷐﷮1 + 𝑥﷯ + ﷐﷮𝑥﷯﷯ . 𝑑𝑥 = ﷐0﷮1﷮﷐﷐﷮1 + 𝑥﷯ + ﷐﷮𝑥﷯﷮﷐﷐﷐﷮1 + 𝑥﷯ ﷯﷮2﷯ − ﷐﷐﷐﷮𝑥﷯ ﷯﷮2﷯﷯﷯ . 𝑑𝑥 = ﷐0﷮1﷮﷐﷐﷮1 + 𝑥﷯ + ﷐﷮𝑥﷯﷮1 + 𝑥 − 𝑥﷯﷯ . 𝑑𝑥 = ﷐0﷮1﷮﷐﷐﷮1 + 𝑥﷯ + ﷐﷮𝑥﷯﷮1﷯﷯ . 𝑑𝑥 = ﷐0﷮1﷮﷐﷮1+𝑥﷯﷯ . 𝑑𝑥+﷐0﷮1﷮﷐﷮𝑥﷯﷯ . 𝑑𝑥 = ﷐0﷮1﷮﷐﷐1+𝑥﷯﷮﷐1﷮2﷯﷯﷯ 𝑑𝑥+﷐0﷮1﷮﷐﷐𝑥﷯﷮﷐1﷮2﷯﷯﷯ 𝑑𝑥 = ﷐﷐﷐﷐﷐1 + 𝑥﷯﷮﷐1﷮2﷯ + 1﷯﷮﷐1﷮2﷯ + 1﷯﷯﷮0﷮1﷯ + ﷐﷐﷐﷐𝑥 ﷮﷐1﷮2﷯ + 1﷯﷮﷐1﷮2﷯ + 1﷯﷯﷮0﷮1﷯ = ﷐﷐﷐﷐﷐1 + 𝑥﷯﷮﷐3﷮2﷯﷯﷮﷐3﷮2﷯﷯﷯﷮0﷮1﷯ + ﷐﷐﷐﷐𝑥 ﷮﷐3﷮2﷯﷯﷮﷐3﷮2﷯﷯﷯﷮0﷮1﷯ = ﷐﷐2﷮3﷯﷐﷐﷐1+𝑥﷯﷮﷐3﷮2﷯﷯﷯﷮0﷮1﷯ + ﷐2﷮3﷯﷐﷐﷐𝑥 ﷮﷐3﷮2﷯﷯﷯﷮0﷮1﷯ = ﷐2﷮3﷯ ﷐﷐﷐1+1﷯﷮﷐3﷮2﷯﷯−﷐﷐1+0﷯﷮﷐3﷮2﷯﷯﷯ + ﷐2﷮3﷯ ﷐﷐﷐1﷯﷮﷐3﷮2﷯﷯−﷐﷐0﷯﷮﷐3﷮2﷯﷯﷯ = ﷐2﷮3﷯ ﷐﷐﷐2﷯﷮﷐3﷮2﷯﷯−﷐﷐1﷯﷮﷐3﷮2﷯﷯﷯ + ﷐2﷮3﷯ ﷐1−0﷯ = ﷐2﷮3﷯ . ﷐﷐2﷯﷮﷐3﷮2﷯﷯−﷐2﷮3﷯ ﷐1﷯+﷐2﷮3﷯ ﷐1﷯ = ﷐2﷮3﷯ ﷐﷐2﷯﷮﷐3﷮2﷯﷯ = ﷐2﷮3﷯ ﷐﷐2﷯﷮﷐3﷮2﷯﷯ = ﷐2﷮3﷯ ﷐﷐﷐﷐2﷯﷮﷐1﷮2﷯﷯﷯﷮3﷯ = ﷐2﷮3﷯ ﷐﷐﷐﷮2﷯ ﷯﷮3﷯ = ﷐2﷮3﷯ . 2 ﷐﷮2﷯ = ﷐𝟒 ﷐﷮𝟐﷯﷮𝟑﷯

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