integrate from 0 to 2: 2x*sqrt(x+2) dx

There are a few ways to go about this question. I will do a substitution using u=(x+2)^1/2. From this equation we need to find x in terms of u, the new limits of the integral and dx in terms of du. By rearranging we see that x=u²-2. Also if we substitute the original limits into the equation u=(x+2)^1/2, we have when x=0, u=2^1/2 and when x=2, u=4. Now by differentiating x=u²-2 with respect to u we get dx/du=2u so dx=2udu. Now we have all our information we can write the integral completely in terms of u as we are now integrating with respect to u. We have the integral from 2^1/2 to 2 of 2(u²-2)u2u du which simplifies to 4u⁴-8u² du. Integrating that gives [4u⁵/5-8u³/3] from 2^1/2 to 2. Now substituting the limits in and simplifying gives the final answer: 32/15(2+2^1/2).