Integral Question

[uberathlete]

Hi everyone. I just had a test and there was one integral I couldn't solve. I tried solving it agian when I got home but I still can't get it. I tried substitution and integration by parts but no dice. The problem is:

intgeral sign (x^3/2)/(x+10) dx

If someone could show me how it's done, it'd be greatly appreciated. Thanks!

 

[Unco]

\(\displaystyle \L\mbox{ \int \frac{x^{\frac{3}{2}}}{x + 10} dx}\)

I recall you've worked with trig subs.

I prefer to cut out the middleman and let \(\displaystyle \mbox{ x = 10\tan^2{(\theta)} \Rightarrow dx = 20\tan{(\theta)}\sec^2{(\theta)} d\theta}\).

 

[soroban]

Hello, uberathlete!

\(\displaystyle \L\int \frac{x^{\frac{3}{2}}}{x\,+\,10}\,dx\)

Since it has a square root:
\(\displaystyle \;\;\text{let: }u\,=\,x^{\frac{1}{2}}\;\;\Rightarrow\;\;x\,=\,u^2\;\;\Rightarrow\;\;dx\,=\,2u\,du\)

Substitute: \(\displaystyle \L\:\int \frac{u^3}{u^2\,+\,10}(2u\,du)\;=\;2\int\frac{u^4}{u^2\,+\,10}\,du\)

Some long division and we have: \(\displaystyle \L\:2\int\left(u^2\,-\,10\,+\,\frac{100}{u^2\,+\,10}\right)\,du\)

Got it?