Question

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

Hint:

Put \(x+2=t^2\)

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]

=\(\frac{16(2+√2)}{15}\)

=\(\frac{16√2(√2+1)}{15}\)