Mathematics

Evaluate : 
    $$I=\int { \left( 2x+3 \right)  } \sqrt { x-1 } dx$$


SOLUTION
$$ I = \int (2x+3)\sqrt{(x-1)}dx$$
$$ = \int (2x-2+5)\sqrt{(x-1)}dx$$
$$ = \int 2(x-1)\sqrt{(x-1)} dx+5 \int \sqrt{(x-1)}dx$$
$$ = 2\int (x-1)^{\frac{3}{2}}dx+5\int (x-1)^{\frac{1}{2}}dx$$
$$ = 2\frac{(x-1)^{\frac{3}{2}+1}}{(\frac{3}{2}+1)}+5\frac{(x-1)^{\frac{1}{2}+1}}{\frac{1}{2}+1} +c$$
$$ I = \frac{4}{5}(x-1)^{^{5/2}}+\frac{10}{3}(x-1)^{\frac{3}{2}}+c$$
$$ \int (2x+3)\sqrt{x-1}dx = (x-1)[\frac{4}{5}(x-1)^{\frac{3}{2}}+\frac{10}{3}\sqrt{x-1}]+c$$
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Subjective Medium Published on 17th 09, 2020
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