# Fundamental theorem of calculus

1. Nov 9, 2014

### nicolauslamsiu

• Member warned about posting with no effort
1. The problem statement, all variables and given/known data
Using Fundamental Theorem of Calculus to find the derivative

2. Relevant equations
upper limit=x^2, lower limit=4x

∫ { 1 / [1+ (sin t)^2] }dt

3. The attempt at a solution
two independent variables are involved, how should i find the derivative?

2. Nov 9, 2014

### 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?

3. Nov 9, 2014

### nicolauslamsiu

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

4. Nov 9, 2014

### haruspex

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.

5. Nov 10, 2014

### 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.