I've been doing integrals non stop preparing for my exam tomorrow, and one has left me stumped for a few days. I've tried coming back to it several times, but I can't seem to manipulate it. It is as follows:
$$\int\frac{x^2}{\sqrt{x+1}}\;dx$$
I've tried different u-sub methods but nothing seems to work for me. My next thought was integration by parts, but when I attempt that method my answer isn't even close to the answer key given by my professor. The answer key shows this as the answer:
$$\frac25(x+1)^\frac{5}{2}-\frac43(x+1)^\frac{3}{2}+2\sqrt{x+1}+C$$
What would be the best technique to handle this integral? I feel like i'm missing something simple, but after 2 days it still hasn't come to me and it's quite frustrating. Thanks in advance.
Answer
Use the substitution method.
Let $u=x+1$, then $u-1=x$ and $du=dx$. Try it.
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