Question

Rationalizing. Evaluate the Integral

Answer 1

\[\int\limits_{0}^{1} x^2\sqrt{8x+6} dx\]

Answer 2

\(\large\color{black}{\displaystyle\int\limits_{~}^{~}x^2\sqrt{8x+6}~dx}\)

u=8x+6 ---> (u-6)/8
du=8 dx ---> (1/8)du = dx

x^2 = (u-6)^2/64

Answer 3

so for this type of problem, i need to find my u and x^2 as well?

Answer 4

you can also try \(u=\sqrt{8x+6}\) if you like

Answer 5

\(\large\color{black}{\displaystyle\frac{1}{64}\int\limits_{~}^{~}(u-6)^2\sqrt{u}~du}\)

Answer 6

yes that u sub was what I offered.

Answer 7

with the square root, but it makes no difference really

Answer 8

\(\large\color{black}{\displaystyle\frac{1}{64}\int\limits_{~}^{~}u^2\sqrt{u}-12\sqrt{u}+36\sqrt{u}{~~}du}\)

Answer 9

your way is probably easier

Answer 10

well, just that integral term by term, then sub the u back after integration....

(shouldn't be hard, just applying u-sub, re-wrote x^2 and dx in terms of u,
later had to apply some basic algebra)

Answer 11

ok thank you! trying out the problem now

Answer 12

how did u = 8x+6 become (u-6)/8?

Answer 13

(u-6)/8 = x

that is the value that I get for x when I rearranged

Answer 14

And thus x^2 became, (u-6)^2/64

Answer 15

oh i see! you were solving for x and got that. then you just squared it to get x^2 and to plug in x^2 into the problem

Answer 16

yes plug in (u-6)^2/64 instead of x^2 into the problem

Answer 17

I might have done sommething wrong though...

Answer 18

ok thanks! lol it looks good to me! its how i do trig substitution with finding the x^2 also

Answer 19

\(\large\color{black}{\displaystyle\int\limits_{~}^{~}x^2\sqrt{8x+6}~dx}\)

u=8x+6 ---> x=(u-6)/8 ---> x^2 = (u-6)^2/64
du=8 dx ---> (1/8)du = dx

\(\large\color{black}{\displaystyle\int\limits_{~}^{~}\frac{(u-6)^2}{64}\sqrt{u}\left(\frac{du}{8}\right)}\)

Answer 20

So, really we get:

\(\large\color{black}{\displaystyle\frac{1}{512}\int\limits_{~}^{~}\left(u^2\sqrt{u}-12u\sqrt{u}+36\sqrt{u}\right){~}du}\)

Answer 21

\(\large\color{black}{\displaystyle\frac{1}{512}\int\limits_{~}^{~}\left(u^{2.5}-12^{1.5}+36u^{0.5}\right){~}du}\)

Answer 22

oh wow that got complicated! lol thank you i am trying what you did

Answer 23

yw

Answer 24

Psst...

Answer 25

lel c: Go ooooon :3

Answer 26

Anyway what was I saying...?

Answer 27

Oh yeah PSST....

Answer 28

MEOW, I AM A DINOSAUR. MEEOW AND RAUUURRRR!!

Answer 29

xD

Answer 30

lol

Answer 31

Lol