Activity 5.2.4

Evaluate each of the following derivatives and definite integrals. Clearly cite whether you use the First or Second FTC in so doing.

1. $\displaystyle \frac{d}{dx} \left[ \int_4^x e^{t^2} \, dt \right]$

2. $\displaystyle \int_{-2}^x \frac{d}{dt} \left[ \frac{t^4}{1+t^4} \right] \, dt$

3. $\displaystyle \frac{d}{dx} \left[ \int_{x}^1 \cos(t^3) \, dt \right]$

4. $\displaystyle \int_{3}^x \frac{d}{dt} \left[ \ln(1+t^2) \right] \, dt$

5. $\displaystyle \frac{d}{dx} \left[ \int_4^{x^3} \sin(t^2) \, dt \right]$