Exercises - Fundamental Theorem of Calculus (Part I)

  1. Evaluate $\displaystyle{\frac{d}{dx} \int_0^x t \sqrt{4-t^3}\,dt}$

    $\displaystyle{x \sqrt{4-x^3}}$

  2. Evaluate $\displaystyle{\frac{d}{d\theta} \int_0^{\sin \theta} \frac{1}{1-x^2} dx}$

    Suppose $\displaystyle{F(\theta) = \int_0^\theta \frac{1}{1-x^2}\,dx}$.

    The we seek $\displaystyle{\frac{d}{dx} F(\sin \theta)}$, the derivative of a composition -- so use the chain rule.

    $\displaystyle{\frac{1}{1-\sin^2 \theta} \cdot \cos \theta = \frac{1}{\cos^2 \theta} \cdot \cos \theta = \sec \theta}$