Fundamental theorem of calculus

[nicolauslamsiu]

Homework Statement

Using Fundamental Theorem of Calculus to find the derivative

2. Homework Equations
upper limit=x^2, lower limit=4x
∫ { 1 / [1+ (sin t)^2] }dt

The Attempt at a Solution

two independent variables are involved, how should i find the derivative? [/B]

 

[FeDeX_LaTeX]

Let F(x) = \int_{4x}^{x^2} \frac{dt}{1 + \sin^{2}t}, and put G(y) := \int_{0}^{y} \frac{dt}{1 + \sin^{2}t}.

Then, F(x) = G(x^2) - G(4x), by the domain splitting property of the Riemann integral. Does this help things?

 

[nicolauslamsiu]

oo) yup... sure... i think i know how to solve now... thanks

 

[haruspex]

two independent variables are involved,

Not really. t is a 'dummy variable' that has no existence outside the integral. The integral as a whole is a function of x only.

 

[TheMathSorcerer]

It's probably worth making an attempt using the first FTC and understanding it.

You could use this,
Set $F(x) = \int_{\alpha(x)}^{\beta(x)}{ f(t) dt}$. Then,

$F'(x) = f(\beta(x))\beta'(x) - f(\alpha(x))\alpha'(x)$

However it's probably worth understanding the first FTC before jumping into this.