Integration with u-subsitution

[Vertciel]

Hello everyone,

I am not getting the right answer for this integration exercise. Could someone please show me where I may have erred?

Thank you.

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1. \(\displaystyle \int_{0}^{3} xe^{x^2} dx\)

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I am uncertain about this, but I let:

\(\displaystyle u = x\) and \(\displaystyle du = dx\).

Therefore, the new limits of integration would still be 3 and 0, as \(\displaystyle u = x\).

Now:

\(\displaystyle \frac{u^2}{2} \frac{e^{u^2 + 1}}{u^2 + 1} \mid_{0}^{3}\)

Integral = \(\displaystyle \frac{9e^{10}}{20}\)

However, this is not the correct answer.

 

[galactus]

All you done was replace x with u. That doesn't do you much good.

Better yet, let \(\displaystyle u=e^{x^{2}}, \;\ du=2xe^{x^{2}}, \;\ \frac{du}{2}=xe^{x^{2}}dx\)

Then we get \(\displaystyle \frac{1}{2}\int_{1}^{e^{9}}du\)

 

[Vertciel]

Thank you for your reply, galactus.

I just want to verify that when the function has finished integration, I would end up with this:

\(\displaystyle \frac{1}{2}\int_{1}^{e^{9}}du\)

\(\displaystyle i(x)= \frac{1}{2} \frac{x^2}{2}ex^2\)

Then at this point, I would just substitute \(\displaystyle i(e^9)\) and \(\displaystyle i(1)\).

 

[galactus]

I do not know what this is

\(\displaystyle i(x)= \frac{1}{2} \frac{x^2}{2}ex^2\),

But yes, just integrate and use your limits. This integral couldn't be any easier.

When you integrate, you get u. Then \(\displaystyle \frac{1}{2}(e^{9}-1)\)

That's it.

 

[Vertciel]

I do not know what this is

\(\displaystyle i(x)= \frac{1}{2} \frac{x^2}{2}ex^2\),

Wouldn't this be the result of me integrating (i.e. after your last step in your previous post)?

 

[galactus]

\(\displaystyle \int xe^{x^{2}}dx=\frac{e^{x^{2}}}{2}\)

 

[DR._Glockman]

Ever notice the more complicated the integral, the easier it is to find an antiderivative which is an elementary function?

For example, try finding a antiderivative which is an elementary function for \(\displaystyle \int e^{x^{2}}dx\).

No can do.