Fundamental Theorem of Calculus, Part 1 practice problems
Differentiate an integral whose upper limit is a function of x. 30 problems with worked solutions; in 30 of them every equation is proved by a computer algebra system.
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{\sqrt{x}} \sqrt{t^{2} + 1} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{\sqrt{x}} e^{- t^{2}} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{\sin{\left(x \right)}} t^{3} \cos{\left(t \right)} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{1}^{x^{3}} \frac{1}{t^{4} + 1} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{x} \frac{1}{t^{4} + 1} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{x^{3}} \ln{\left(t^{2} + 2 \right)} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{1}^{x^{2}} \frac{1}{t^{4} + 1} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{1}^{x} \sqrt{t^{2} + 2} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{1}^{\sin{\left(x \right)}} \ln{\left(t^{2} + 2 \right)} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{x} \ln{\left(t^{2} + 2 \right)} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{x} \sin{\left(t^{2} \right)} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{x^{3}} \ln{\left(t^{2} + 2 \right)} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{1}^{\sqrt{x}} \sqrt{t^{2} + 4} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{x} e^{- t^{2}} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{\sin{\left(x \right)}} e^{- t^{2}} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{x^{3}} e^{- t^{2}} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{3 x} \sin{\left(t^{2} \right)} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{1}^{3 x} t^{3} \cos{\left(t \right)} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{1}^{\sqrt{x}} e^{- t^{2}} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{3 x} \sin{\left(t^{2} \right)} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{\sin{\left(x \right)}} \frac{1}{t^{4} + 1} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{x^{2}} e^{- t^{2}} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{3 x} \frac{1}{t^{4} + 1} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{x^{3}} \sqrt{t^{2} + 1} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{x^{3}} \sqrt{t^{2} + 5} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{\sqrt{x}} e^{- t^{2}} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{\sin{\left(x \right)}} \frac{1}{t^{4} + 1} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{0}^{x^{2}} \sqrt{t^{2} + 2} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{2}^{x} \sqrt{t^{2} + 1} \, dt \).
Find \( \displaystyle \dfrac{d}{dx} \displaystyle \int_{1}^{x^{3}} \sin{\left(t^{2} \right)} \, dt \).