Question

find formulas for the function represented by the integral

Answer 1

\[\int\limits_{2}^{x}u^4du\]

Answer 2

We know the basic form for integrating gives us:
\[\int\limits\limits_{}^{}x^ndx=\frac{ x ^{n+1} }{ n+1 }\]
So what is our implicit integral of u^4 du?

Answer 3

u^5/5+c

Answer 4

Perfect. Then we apply the Fundamental Theorem of Calculus:
\[\int\limits_{a}^{b}xdx=F\]
\[F(b)-F(a)\]

Answer 5

\[16/5-x^5/5+c??\]

Answer 6

Close. Really close. Remember to do your upper bound first:
\[(\frac{ x^5 }{ 5 }+c)-(\frac{2^5}{5}+c)\]

Answer 7

so i have to add a constant to both factors

Answer 8

Yes, but usually given bounds (in this case x and 2) the c's are not accounted for and "cancel" persay. You're finding the area under the curve between 2 and x.

Answer 9

So what would our expression simplify to?

Answer 10

\[x^5/5-32/5\]

Answer 11

Yep, exactly. Questions regarding these types of problems?

Answer 12

i understand these now Im struggling with the ones that use sec and e to an exponent

Answer 13

That's understandable. Integrals of Trig functions and e can be tricky.

Answer 14

\[\int\limits_{x}^{0}e^(-t)dt\]

Answer 15

e^(-t)?

Answer 16

yeah sorry it came out wierd

Answer 17

No worries. That one is kind of tricky, but if you do substitution where u=-t that should help, hopefully...
\[u=-t\]
\[du=-dx\]
\[\int\limits e^u(-du)= - \int\limits e^udu\]

Answer 18

sorry, should be dt, not dx. Habit. :D

Answer 19

ha its fine and then from there I am able to apply the limit?

Answer 20

yeah, integral of e^x is just e^x+c so integral of e^(-t) is just -e(-t)+c. Then apply the FTC we talked about earlier: F(b)-F(a) giving us:
\[(-e^{-0}+c)-(-e^{-x}+c)\]

Answer 21

1+e^-x

Answer 22

Yep, you got it! :D

Answer 23

yayyy do you know how to express the antiderivative?

Answer 24

-e(-t)+c

Answer 25

\[f(x)=secx F(0)=0\]

Answer 26

Integral of sec(x)?