BC Calculus FTC Worksheet

Find the following:

1. If \( F(x) = \displaystyle \int_{1}^{x} \frac{1}{1+t^2} \, dt \), find \(F'(x) \).
2. If \( F(x) = \displaystyle \int_{3}^{2x}\sqrt{t^2 + 1} \, dt\), find \(F'(x)\).
3. If \( F(x) = \displaystyle \int_{x}^{2x} \frac{1}{t} \, dt\) for \(x > 0\), find \(F'(x)\).
4. If \( F(x) = \displaystyle \int_{0}^{\sqrt{x}} \sin{ \left(t^2 \right) } dt\), find \(F'(x)\).
5. If \( x > 0\), then find \( \displaystyle \frac{d}{dx} \int_{1}^{\frac{1}{x}} \frac{2}{t} \ dt \).
6. Find \( \displaystyle \int_{1}^{4} \frac{d}{dx} \sqrt{x^2 - 1} \, dx\).