Question

Evaluate the integral: $\int^2_0 x\sqrt{x+2}$

Answer 1

$\int_{0}^{2} \sqrt{x+2} \,dx$

Let x+2=t2⧠dx=2tdt

When x=0,t=√2 and,when x=2,t=2

∴$\int_{0}^{2} \sqrt{x+2} \,dx$ =∫2√2(t2-2)√t22tdt

=2∫2√2(t2-2)t2dt

=2∫2√2(t4-2t2)dt

=2[t5/5-2t3/3]2√2

=2[$\frac{32}{5}-\frac{16}{3}-\frac{4√2}{5}+\frac{4√2}{3}$]

=2[$\frac{96-80-12√2+20√2}{15}$]

=2[16+8√2/15]

=$\frac{16(2+√2)}{15}$

=$\frac{16√2(√2+1)}{15}$