Question

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

Answer 1

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

$=\int (2x-2+5)\sqrt{(x-1)}dx$

$=\int 2(x-1)\sqrt{(x-1)}dx+5\int \sqrt{(x-1)}dx$

$=2\int (x-1{)}^{\frac{3}{2}}dx+5\int (x-1{)}^{\frac{1}{2}}dx$

$=2\frac{(x-1{)}^{\frac{3}{2}+1}}{(\frac{3}{2}+1)}+5\frac{(x-1{)}^{\frac{1}{2}+1}}{\frac{1}{2}+1}+c$

$I=\frac{4}{5}(x-1{)}^{{}^{5/2}}+\frac{10}{3}(x-1{)}^{\frac{3}{2}}+c$

$\int (2x+3)\sqrt{x-1}dx=(x-1)\left[\frac{4}{5}\right(x-1{)}^{\frac{3}{2}}+\frac{10}{3}\sqrt{x-1}]+c$