Question

∫ square root of 1+3x^3 (x^2)dx

Answer 1

u- sub for this one. put
\[u=1+3x^3, du=9x^2dx,\frac{1}{9}du=x^2\] and then integrate
\[\frac{1}{9}\int\sqrt{u}du\] using the power rule backwards

Answer 2

so is that the final answer?

Answer 3

oh no you still have to find the "anti - derivative"

Answer 4

totally lost

Answer 5

you are looking for the function whose derivative is
\[\sqrt{1+3x^3}x^2\]

Answer 6

correct

Answer 7

you are going to run the chain rule backwards, because this looks almost like a composite function times its derivative

Answer 8

yes it is

Answer 9

so you have to figure out how to adjust so that the constants work out correctly. that is the point of the u - sub

Answer 10

do you know how to find the "anti - derivative" of
\[\sqrt{u}\]?

Answer 11

u^1/2

Answer 12

that is another name for
\[\sqrt{u}\] it is
\[\sqrt{u}=u^{\frac{1}{2}}\] you still need the anti derivative

Answer 13

you have to add one to the exponent and then divide by that number

Answer 14

3/2u^3/2

Answer 15

right. exactly!

Answer 16

I shocked myself

Answer 17

so now we have changed the original integral to
\[\frac{1}{9}\int\sqrt{u}du\] and the anti derivative will be
\[\frac{1}{9}\times \frac{2}{3}u^{\frac{3}{2}}\]

Answer 18

2/27

Answer 19

oh actually you made a slight mistake. dividing by 3/2 is the same as multiplying by 2/3, so it should have been
\[\frac{2}{3}u^{\frac{3}{2}}\]

Answer 20

oh yeah
forgot to flip

Answer 21

in any case you get
\[\frac{2}{27}u^{\frac{3}{2}}\] and now replace u by
\[1+3x^3\] to get
\[\frac{2}{27}(1+3x^3)^{\frac{3}{2}}\]

Answer 22

or if you prefer
\[\frac{2}{27}\sqrt{1+3x^3}^3+C\]

Answer 23

what about the x^2

Answer 24

you can check your answer by taking the derivative and seeing that you will get
\[\sqrt{1+3x^3}\times x^2\]

Answer 25

the chain rule will give it to you . that was the idea behind the original substitution, making
\[u=1+3x^3\]

Answer 26

ummmm???

Answer 27

let us do a simpler example. what is the derivative of
\[\sin(x^2)\]?

Answer 28

OH I get it that's wehre the 1/9 came from

Answer 29

righto!~

Answer 30

duh...okay thank you fro helpng me to the answer 2/27 SQRT(1+3x^3 +C

Answer 31

careful it is
\[\frac{2}{27}\sqrt{1+3x^3}^3+C\]

Answer 32

don't forget the "cube" because you are raising to the power of 3/2

Answer 33

got it!

Answer 34

Can you walk me through another one?
lim e^3x-1/4x
x->0