Fundemental Theorem Questions (I think)

• Aug 1st 2008, 03:58 AM
skirk34
Fundemental Theorem Questions (I think)
For maths, i dont know how to work out the following:

1)Find the value of;
Integral(0,4): f'(x) ( (f(x))^3 -2f(x) -3) dx
given that f(0) = 2, f'(0) = -6, f(4) = 3 and f'(4) = -3

2) y= Integral(5,5x^4):sin( t^2 - 5 ) dt
whats y'?

(I'm new so sorry, i dont know how to write in equation form)
• Aug 1st 2008, 04:19 AM
wingless
Hint:

$\displaystyle \int f'(x) [f(x)]^n~dx$

Let $\displaystyle u = f(x)$, then $\displaystyle du = f'(x)~dx$.

$\displaystyle \int u^n~du$

- If $\displaystyle n\neq -1$, then it's $\displaystyle \frac{u^{n+1}}{n+1}= \frac{[f(x)]^{n+1}}{n+1}$.

- If $\displaystyle n = -1$, then it's $\displaystyle \ln u = \ln f(x)$
• Aug 1st 2008, 04:24 AM
kalagota
Quote:

Originally Posted by skirk34
For maths, i dont know how to work out the following:

1)Find the value of;
Integral(0,4): f'(x) ( (f(x))^3 -2f(x) -3) dx
given that f(0) = 2, f'(0) = -6, f(4) = 3 and f'(4) = -3

set $\displaystyle u = f(x)$
$\displaystyle du = f'(x)dx$

at $\displaystyle x=0: u=f(0)=2$
at $\displaystyle x=4: u=f(4)=3$

thus

$\displaystyle \int_2^3 (u^3 -2u -3)du = ...$
• Aug 1st 2008, 04:28 AM
kalagota
Quote:

Originally Posted by skirk34
For maths, i dont know how to work out the following:

2) y= Integral(5,5x^4):sin( t^2 - 5 ) dt
whats y'?

(I'm new so sorry, i dont know how to write in equation form)

if $\displaystyle F(x) = \int_a^{f(x)} h(t)dt$ where $\displaystyle a$ is a constant,

then $\displaystyle F'(x) = h(f(x)) \, f'(x)$
• Aug 1st 2008, 05:04 AM
skirk34
For the second question I got the answer:
y'(x) = sin(x^2 - 5).20x^3

The quiz its from isn't liking my answer though (Crying)
Heres the hint it gives me:
Let the integrand be f(t) and let an antiderivative of f(t) be F(t). Then y(x) = F(5 x^4) - F(5), where F(5) is just a constant.
• Aug 1st 2008, 05:09 AM
kalagota
Quote:

Originally Posted by skirk34
For the second question I got the answer:
y'(x) = sin(x^2 - 5).20x^3

The quiz its from isn't liking my answer though (Crying)
Heres the hint it gives me:
Let the integrand be f(t) and let an antiderivative of f(t) be F(t). Then y(x) = F(5 x^4) - F(5), where F(5) is just a constant.

im sorry.. i edited my last post.. take a look at it..