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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS1 Mark
Question
Choose the correct answer in Exercise.

$\int\sqrt{\text{x}^2-8\text{x}+7}\text{dx}$ is equal to

  1. $\frac{1}{2}(\text{x}-4)\sqrt{\text{x}^2-8\text{x}+7}+9\text{log}\Bigg|\text{x}-4+\sqrt{\text{x}^2-8\text{x}+7}\Bigg|+\text{C}$

  2.  $\frac{1}{2}(\text{x}+4)\sqrt{\text{x}^2-8\text{x}+7}+9\text{log}\Bigg|\text{x}+4+\sqrt{\text{x}^2-8\text{x}+7}\Bigg|+\text{C}$

  3. $\frac{1}{2}(\text{x}-4)\sqrt{\text{x}^2-8\text{x}+7}-3\sqrt2\text{log}\Bigg|\text{x}-4+\sqrt{\text{x}^2-8\text{x}+7}\Bigg|+\text{C}$

  4. $\frac{1}{2}(\text{x}-4)\sqrt{\text{x}^2-8\text{x}+7}-\frac{9}{2}\text{log}\Bigg|\text{x}-4+\sqrt{\text{x}^2-8\text{x}+7}\Bigg|+\text{C}$

✓

Answer

  1. $\frac{1}{2}(\text{x}-4)\sqrt{\text{x}^2-8\text{x}+7}+9\text{log}\Bigg|\text{x}-4+\sqrt{\text{x}^2-8\text{x}+7}\Bigg|+\text{C}$

$=\Bigg(\frac{\text{x}-4}{2}\Bigg)\sqrt{(\text{x}-4)^2-3^2}-\frac{3^2}{2}\text{log}\Big|\text{x}-4+\sqrt{(\text{x}-4)^2-3^2}\Big|+\text{C}$​​​​​​​

$\Bigg[\therefore\ \int\sqrt{\text{x}^2-\text{a}^2}\text{dx}=\frac{\text{x}}{2}\sqrt{\text{x}^2-\text{a}^2}-\frac{\text{a}^2}{2}\text{log}\Big|\text{x}+\sqrt{\text{x}^2-\text{a}^2}\Big|\Bigg]$

$=\Bigg(\frac{\text{x}-4}{2}\Bigg)\sqrt{\text{x}^2-8\text{x}+7}-\frac{9}{2}\text{log}\Big|\text{x}-4+\sqrt{\text{x}^2-8​​\text{x}+7}\Big|+\text{C}$

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