Question

Guess an antiderivative for the integrand function and Validate your guess by differentiation and then evaluate the given definite integral.
(a) integral of xe^(x^2)dx, from 0 to 1
(b) inegral of (x dx/ sqrt (1+x^2)), from 2 to 5

Answer 1

what are your guesses?

Answer 2

I don't have any. I don't understand where to start.

Answer 3

you start with an assumption. use the derivative rules that you were taught to consider a function that could possibly derive into these.

Answer 4

there is only one rule that i can think of that pertains to an "e"

and there are about 2 off hand that i can think of that derive to a sqrt on the bottom

Answer 5

so is the antiderivative of ex^(x^2) just ex^(x^2)?

Answer 6

well, we would check the assumptions by actually using the derivative rules:

one rule is: e^u derives to u' e^u; so lets assume e^(x^2) and take the derivative to see if we need to adjust it with come constant

Answer 7

so the derivative of e^(x^2) is...2x(e^(x^2)) right?

Answer 8

correct, and that compares to our setup in that it has that 2 attached to it ... lets use some arbitrary constant k like this:\[\large k~e^{x^2}\to2k~xe^{x^2}\]
such that 2k = 1. what does the value of k need to be?

Answer 9

k= 1/2

Answer 10

good, then we know the antiderivative of x e^(x^2) = 1/2 e^(x^2)

Answer 11

wait..but what happened to "x" in from of "e"?

Answer 12

we are finding derivatives and anti derivatives ....

the derivative of 1/2 e^(x^2) is x e^(x^2)

therefore, the antiderivative of x e^(x^2) is 1/2 e^(x^2)

we are performing an operation on one function to create another function

Answer 13

Oh! ok i see now

Answer 14

do you know how to evaluate an integral over an interval?

Answer 15

umm no

Answer 16

it pretty simple

spose we have two functions so that: f derives to g

the integral of g, over an interval a to b can be formulated by using
its anti derivative f: f(b)-f(a)

Answer 17

ok..so then it would be \[\frac{ 1 }{ 2 }e ^{x ^{2}}]\left(\begin{matrix}1 \\ 0\end{matrix}\right)= \frac{ 1 }{ 2 }e ^{1 ^{2}}-\frac{ 1 }{ 2 }e ^{0 ^{2}}\]?

Answer 18

very good :)

Answer 19

so would that be 0- 1/2 ?

Answer 20

which is -1/2

Answer 21

no\[\frac12e^1-\frac12e^0\]
\[\frac12e-\frac12(1)\]
\[\frac12e-\frac12\]
\[\frac12(e-1)\]

Answer 22

ok gotcha

Answer 23

the other one is similar to this one, the key is in finding a suitable anti derivative

what would be your guess of a function that derives to look something like: 1/sqrt(x)

Answer 24

maybe something function that derives to something looking like:\[\frac{u'}{\sqrt{u}}\]