Need help understanding Fundamental theorm

**swatpup32** · April 15th, 2009, 10:53 PM

Why is the Integral
$\frac{d}{dx}\int_1^{sinx} tant dt$
equal to

$tan(sinx) \cdot cosx$

Why don't you have to take the the antiderivative of tan(t). Like you would if it was
$\frac{d}{dx}\int_1^{sinx} t+5$

Also why is the lower limit not used when solving the problem?

**Grandad** · April 16th, 2009, 12:41 AM

Hello swatpup32

Originally Posted by swatpup32

Why is the Integral
$\frac{d}{dx}\int_1^{sinx} tant dt$
equal to

$tan(sinx) \cdot cosx$

Why don't you have to take the the antiderivative of tan(t). Like you would if it was
$\frac{d}{dx}\int_1^{sinx} t+5$

Also why is the lower limit not used when solving the problem?

It doesn't matter what the function of $t$ is, the explanation goes like this:

$\frac{d}{dx}\int_1^{\sin x} f'(t)\,dt = \frac{d}{dx}\Big[f(t)\Big]_1^{\sin x}$

$= \frac{d}{dx}\Big(f(\sin x) - f(1)\Big)$

$= f'(\sin x) \times \frac{d}{dx}(\sin x) - 0$

$=f'(\sin x) \cos x$

OK?

Grandad

**m3th0dman** · April 16th, 2009, 05:01 AM

Originally Posted by swatpup32

Why is the Integral
$\frac{d}{dx}\int_1^{sinx} tant dt$
equal to

$tan(sinx) \cdot cosx$

Why don't you have to take the the antiderivative of tan(t). Like you would if it was
$\frac{d}{dx}\int_1^{sinx} t+5$

Also why is the lower limit not used when solving the problem?

You use the Fundamental Theorem of Calculus:
$ \int_{a}^{x}f(t)dt=F(x) $
and F'(x)=f(x)

so you have
$ (\int_{a}^{g(x)}f(t)dt)'=F(g(x))g'(x) $

For the question with the lower limit, you have the formula, from Leibinz-Newton formula:
$ \int_{a}^{b}f(x)dx=F(b)-F(a)=-(F(a)-F(b))=-\int_{b}^{a}f(x)dx $

Hope you are familiar with Lagrange's notation.
$ f'(x)=\frac{df}{dx} $

Math Help Forum: Need help understanding Fundamental theorm

LinkBack

Thread Tools

Display

Need help understanding Fundamental theorm

Similar Math Help Forum Discussions

Fundamental Theorm

Fundamental theorm of calculus

Understanding First Fundamental Theorum of Calculus

Proof for Part 2 of the fundamental theorm

Fundamental Theorm of Calculus