Question

$\int \sqrt{x}.\log xdx=$

• A. $\frac{2}{3}{x}^{3/2}.\log x-\frac{4}{9}{x}^{3/2}+c$
• B. $\frac{2}{3}{x}^{3/2}.\log x+x^{3/2}+c$
• C. $x^{3/2}.(\log x-\frac{2}{3})+c$
• D. $\frac{2}{5}{x}^{3/2}(\log x+1)+c$

Answer 1

The correct option is A $\frac{2}{3}{x}^{3/2}.\log x-\frac{4}{9}{x}^{3/2}+c$

$\int \sqrt{x}logxdx$
$=logx\int \sqrt{x}dx$ $-\int 1/x(\int \sqrt{x}dx)dx$
$=2/3logxx{}^{3/2}-2/3\int \frac{x^{3/2}}{x}dx$
$2/3logx\left(x{}^{3/2}\right)-2/3\int x{}^{1/2}dx$
$2/3logx\left(x{}^{3/2}\right)-4/9x{}^{3/2}+c$
$2/3x^{3/2}logx4/9x^{3/2}+c$
$\int \sqrt{x}logxdx=2/3x^{3/2}logx-4/9x^{3/2}+c$