Problems

01

Fundamental Theorem of Calculus

Compute the derivative: $$\frac{d}{dx}{\int }_{\ln x}^1\arctan \left(e^t\right)dt$$

02

Indefinite and definite integrals using trig and exp

Compute the following integrals:

• (a) $\int \cot tdt$ (Hint: $\cot t=\frac{\cos t}{\sin t}$.)
• (b) ${\int }_0^{\ln 2}\frac{e^x-e^{-x}}{2}dx$
• (c) ${\int }_{\pi /4}^{\pi /3}{\sec }^2(x)dx$

03

Indefinite and definite integrals using substitutions

Evaluate the following integrals:

• (a) $\int \frac{(\ln x{)}^2}{x}dx$
• (b) ${\int }_0^{1}x{e}^{-x^2}dx$
• (c) $\int \frac{x}{\sqrt{x^2-1}}dx$

04

Indefinite and definite integrals using trig

Compute the following integrals:

• (a) $\int \frac{x}{x+1}dx$ (Hint: set $u=x+1$.)
• (b) ${\int }_{-1/2}^{\sqrt{3}/2}\frac{dx}{\sqrt{1-x^2}}$

05

Indefinite integrals with substitutions

Find the following indefinite integrals:

• (a) $\int x^4{\left(x^5+6\right)}^{9}dx$
• (b) $\int \frac{1}{5-3x}dx$

06

Definite integrals with substitutions

Evaluate the following integrals:

• (a) ${\int }_{-1}^{1/2}\frac{\arcsin (x)}{\sqrt{1-x^2}}dx$
• (b) ${\int }_1^{2}x\sqrt{x-1}dx$ (Hint: set $u=x-1$.)

07

Net Change

Traffic is flowing past a sensor in a freeway at a rate given by: $$V(t)=-300t^2+2000t+3000$$ (The rate is measured in units of cars per hour.)

How many cars pass the sensor during the time window $t\in [0,2]$?