Question

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

• A. $\frac{2}{5}(x+1{)}^{3/2}(3x+8)+c$
• B. $\frac{2}{15}(x-1{)}^{3/2}(3x-8)+c$
• C. $\frac{2}{15}(x+1{)}^{3/2}(3x+8)+c$
• D. $\frac{2}{15}(x-1{)}^{3/2}(3x+8)+c$

Answer 1

The correct option is C $\frac{2}{15}(x+1{)}^{3/2}(3x+8)+c$

$\int (x+2)\sqrt{x+1}dx$
$\int \left[\right(x+1)+1]\left(\sqrt{x+1}\right)dx$
$\int (x+1{)}^{\frac{3}{2}}dx+\int \sqrt{x+1}dx$
$=\frac{2}{5}(x+1{)}^{5/2}+\frac{2}{3}(x+1{)}^{3/2}+c$
$=\frac{2}{15}\left[3\right(x+1{)}^{5/2}+5(x+1{)}^{3/2}]+c$
$=\frac{2}{15}(x+1{)}^{3/2}(3x+8)+c$