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INTEGRALS — Maths STD 12 Science — Question

Gujarat BoardEnglish MediumSTD 12 ScienceMathsINTEGRALS1 Mark
Question
Find the following integrals in Exercises:

$\int(1-\text{x})\sqrt{\text{x}}\text{ dx}$

 

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Answer

$\int(1-\text{x})\sqrt{\text{x }}\text{ dx} =\int\big(\sqrt{\text{x}}-\text{x}\sqrt{\text{x}}\big)\text{ dx}$
$=\int\bigg(\text{x}^\frac{1}{2}-\text{x}^\frac{3}{2}\bigg)\text{ dx}=\int\text{x}^\frac{1}{2}\text{ dx}-\int\text{x}^\frac{3}{2}\text{ dx} $
$=\frac{\text{x}^{\frac{1}{2}+1}}{\frac{1}{2}+1}-\frac{\text{x}^{\frac{3}{2}+1}}{\frac{3}{2}+1}+\text{c} =\frac{\text{x}^{\frac{3}{2}}}{\frac{3}{2}}+ \frac{\text{x}^{\frac{5}{2}}}{\frac{5}{2}}$
$=\frac{2}{3}\text{x}^\frac{3}{2}-\frac{2}{5}\text{x}^{\frac{5}{2}}+\text{c}$

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