I'm currently studying for my calc exam, and i've stumbled across a rather odd problem(at least for me). I've been doing integrals non stop for 2 weeks now and I haven't seen anything like this, so I would like to know if my approach is correct. I feel like i'm not doing it correctly since my answer conflicts with my professor's answer key. It is as follows:
$$\int\sqrt{x^4+x^7}\;dx\;\; or \int(x^4+x^7)^\frac{1}{2}\;dx$$
Since i've been doing(mostly) complex trig sub and integration by parts, I didn't immediately know what to do. I decided to convert the integral to $\int(x^4+x^7)^1/2$ and multiply the exponents:
$$\int\sqrt{x^4+x^7} = \int(x^2+x^\frac{7}{2})\;dx$$
Then I use basic integration to yield:
$$\frac13x^3+\frac29x^\frac{9}{2}+\;C$$
Taking the derivative to check:
$$\frac {d}{dx}(\frac 13x^3 + \frac29x^\frac{9}{2}) = x^2+x^\frac{7}{2}$$
Seems to give me what I started with, but my answer key has this as the answer: $$\frac29(1+x^3)^\frac32+C$$
I can see some similarities to my answer, but it makes me feel like I made a mistake in my technique. Symbolab isn't capable of computing this integral for some reason, and WolframAlpha gives an answer far, far different then either of the integrals I(or my professor) has. Any input would be greatly appreciated as I just want to be as prepare for my exam as much as possible. Thanks!
Answer
HINT
Take out $x^4$ common from squareroot and then put $x^3=t$
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