ASAP Calculus Integration help (by parts and trig)

**valkyrie** · February 11th 2009, 01:28 PM

Hi guys, I'm new on the forum and need a little assistance... we were given this to integrate:

/int x^3/sqrt (x^2+1)dx

And were instructed to integrate by trig integration and integration by parts and then show that they are the same. I got the answer with trig integration:

1/3 sqrt (x^2+1)^3 - 1/sqrt (x^2+1) + c

but I can't figure out how to do it by parts. What should I assign as u and dv and why?

**skeeter** · February 11th 2009, 02:01 PM

/int x^3/sqrt (x^2+1)dx

$\int \frac{x^3}{\sqrt{x^2+1}} \, dx$

integration by parts ...

$u = x^2$

$dv = \frac{x}{\sqrt{x^2+1}} \, dx $

$du = 2x \, dx$

$v = \sqrt{x^2+1}$

$x^2 \sqrt{x^2+1} - \int 2x\sqrt{x^2+1} \, dx $

$x^2\sqrt{x^2+1} - \frac{2}{3}\sqrt{(x^2+1)^3} + C $

$\sqrt{x^2+1}\left[x^2 - \frac{2}{3}(x^2+1)\right] + C$

$\sqrt{x^2+1} \left(\frac{x^2-2}{3}\right) + C$

**Soroban** · February 11th 2009, 02:03 PM

Hello, valkyrie!

It can be done with just by-parts . . .

$\int \frac{x^3}{\sqrt{x^2+1}}\,dx$

. . $\begin{array}{ccccccc}u &=& x^2 & & dv &=&(x^2+1)^{-\frac{1}{2}}(x\,dx) \\ du &=& 2x\,dx & & v &=& (x^2+1)^{\frac{1}{2}} \end{array}$

And we have: . $x^2\sqrt{x^2+1} - \int2x(x^2+1)^{\frac{1}{2}}\,dx$

Got it?

Edit: Skeeter beat me to it . . . *sigh*
.

**skeeter** · February 11th 2009, 03:18 PM

$\int \frac{x^3}{\sqrt{x^2+1}} \, dx$

trig integration ...

$x = \tan{t}$

$dx = \sec^2{t} \, dt$

$\int \frac{\tan^3{t}}{\sqrt{\tan^2{t}+1}} \cdot \sec^2{t} \, dt$

$\int \tan^3{t}\cdot \sec{t} \, dt$

$\int \tan^2{t} \cdot \sec{t}\tan{t} \, dt$

$\int (\sec^2{t} - 1)\sec{t}\tan{t} \, dt$

let $u = \sec{t}$

$\int (u^2 - 1) \, du$

$\frac{u^3}{3} - u + C$

$\frac{\sec^3{t}}{3} - \sec{t} + C$

$\sec{t}\left(\frac{\sec^2{t} - 1}{3} - \frac{2}{3}\right) + C$

$\sqrt{\tan^2{t}+1} \left(\frac{\tan^2{t} - 2}{3}\right) + C$

$\sqrt{x^2 + 1} \left(\frac{x^2-2}{3}\right) + C$

**valkyrie** · February 11th 2009, 04:15 PM

Originally Posted by skeeter

$\int \frac{x^3}{\sqrt{x^2+1}} \, dx$

integration by parts ...

$u = x^2$

$dv = \frac{x}{\sqrt{x^2+1}} \, dx $

$du = 2x \, dx$

$v = \sqrt{x^2+1}$

$x^2 \sqrt{x^2+1} - \int 2x\sqrt{x^2+1} \, dx $

$x^2\sqrt{x^2+1} - \frac{2}{3}\sqrt{(x^2+1)^3} + C $

$\sqrt{x^2+1}\left[x^2 - \frac{2}{3}(x^2+1)\right] + C$

$\sqrt{x^2+1} \left(\frac{x^2-2}{3}\right) + C$

im confused, where does the 2x go from insude the integral?

**Jhevon** · February 11th 2009, 04:21 PM

Originally Posted by valkyrie

im confused, where does the 2x go from insude the integral?

do the integral, you will see where the 2x went (use integration by substitution).

for your general edification, note that the integral can be done using regular substitution, $u = x^2 + 1$ transforms it into $\frac 12 \int (u^{1/2} - u^{-1/2})~du$

**valkyrie** · February 11th 2009, 07:13 PM

thanks a lot! this was really helpful!

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