Find Integral Using Trigonometric Substitutions.. Please Help!!

[shotdoctor]

I've been at this question for a couple hours now, no exaggeration.

Evaluate the integral using the indicated trigonometric substitution. (Use C for the constant of integration.)

. . . . .\(\displaystyle \displaystyle{ \int \,}\) \(\displaystyle \dfrac{x^3}{\sqrt{x^2\, +\, 100\,}}\, dx,\, \). . .\(\displaystyle x\, =\, 10\, \tan\left(\theta\right)\)

I feel like I have the gist down and I know how to get rid of the radical and stuff but I can't seem to get the right answer. Some help would be greatly appreciated. Thanks.

 

[soroban]

Hello, shotdoctor!

\(\displaystyle \displaystyle\int \frac{x^3}{\sqrt{x^2+100}}\,dx\)

Let \(\displaystyle x \,=\,10\tan\theta \quad\Rightarrow\quad dx \,=\,10\sec^2\!\theta\,dx \quad\Rightarrow\quad \sqrt{x^2+100}\,=\,10\sec\theta \)

Substitute: \(\displaystyle \displaystyle\;\int \frac{1000\tan^3\!\theta}{10\sec \theta}(10\sec^2\!\theta\,d\theta) \;=\;1000\int\tan^3\!\theta\sec\theta\,d\theta \)

\(\displaystyle \displaystyle\quad=\; 1000\int\tan^2\!\theta(\sec\theta\tan\theta\, d\theta) \;=\;1000\int(\sec^2\!\theta -1)(\sec\theta\tan\theta\,d\theta) \)

Let \(\displaystyle u \,=\,\sec\theta \quad\Rightarrow\quad du \,=\,\sec\theta\tan\theta\,d\theta\)

Substitute: \(\displaystyle \:\displaystyle 1000\int (u^2-1)\,du\)

Can you finish it now?