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I try to evaluate this two integrals, but I don't know how to proceed:

i) $\int \sqrt{x\sqrt{2x}} dx = \int {2^{\frac{1}{4}}\cdot x^{\frac{3}{4}}}$

ii) $ \int 3^x e^x dx$

What's the best way to evaluate them? Substitution or Intergration by parts?

Any hints are appreciated.

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    You don't solve integrals, you evaluate them.2012-09-12

1 Answers 1

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You’ll do better with the first one if you correct the algebra:

$\sqrt{x\sqrt{2x}}=\left(x(2x)^{1/2}\right)^{1/2}=\left(x\cdot2^{1/2}x^{1/2}\right)^{1/2}=\left(2^{1/2}x^{3/2}\right)^{1/2}=2^{1/4}x^{3/4}\;.$

Now you have $\displaystyle\int2^{1/4}x^{3/4}~dx=2^{1/4}\int x^{3/4}~dx$, which is just a power rule integration.

In the second problem, use the fact that $3^xe^x=(3e)^x$; I’m sure that you’ve been shown how to integrate $a^x$ for a constant $a$.

You don’t need any special techniques for either of them.

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    Thanks a lot, I edited the mistake. And yes, I know how to integrate $a^x$. Thanks for the answer.2012-09-12