1. Integration Problems

Maybe because it's getting late in the evening but I am lost as to how to begin these three problems:

1. Find f(x). f'(x) = e^x - sin(x), f(0) = 1.

2. Find s(t). s''(t) = -1, s'(0) = 3, s(0) = 7.

3. The integral with lower limit 1 and upper limit e, (3/x)dx

Could someone help me get these started? Thanks!

2. Originally Posted by cdx9
Maybe because it's getting late in the evening but I am lost as to how to begin these three problems:

1. Find f(x). f'(x) = e^x - sin(x), f(0) = 1.
$f(x) = \int f'(x)~dx$

your expression for $f(x)$ will have an arbitrary constant $C$. you can find this by plugging in $x = 0$ and setting the expression to 1, since $f(0) = 1$.

2. Find s(t). s''(t) = -1, s'(0) = 3, s(0) = 7.
similar to the last problem, you have to do the above twice.

first find $s'(t) = \int s''(t)~dt$ and find the arbitrary constant, then find $s(t) = \int s'(t)~dt$ and find the arbitrary constant

3. The integral with lower limit 1 and upper limit e, (3/x)dx

Could someone help me get these started? Thanks!
by the fundamental theorem of calculus, if $F(x)$ is the anit-derivative of $f(x)$, then

$\int_a^b f(x) = F(b) - F(a)$

i suppose you can find $\int \frac 3x ~dx$

3. Thanks for the quick reply Jhevon.

I believe I've figured out the first two problems, here is what I came up with for solutions:

1. f(x) = e^x + cos(x) - 1

2. s(t) = (-t^2 / 2) +3t + 7

Feel free to correct me if those are incorrect. The third problem still eludes me. I understand the Fundamental Theorem of Calculus but the "e" in the upper limit is something I don't know how to compute. The problems/examples in my book involving e deal with indefinite integrals or definite integrals with integer values.

4. Originally Posted by cdx9
Thanks for the quick reply Jhevon.

I believe I've figured out the first two problems, here is what I came up with for solutions:

1. f(x) = e^x + cos(x) - 1

2. s(t) = (-t^2 / 2) +3t + 7
both are correct. good job

Feel free to correct me if those are incorrect. The third problem still eludes me. I understand the Fundamental Theorem of Calculus but the "e" in the upper limit is something I don't know how to compute. The problems/examples in my book involving e deal with indefinite integrals or definite integrals with integer values.
$e$ is a mathematical constant, like $\pi$, it is not something you compute, it has a predetermined value. it is on your calculator and everything.

note in particular though, that $\ln x$ means $\log_e x$