Problems

01

Integral computations - trig and exponential

Compute the following indefinite integrals. Don’t forget to add an indefinite constant $C$ to your antiderivative functions.

• (a) $\int \frac{1}{1+x^2}-\frac{1}{\sqrt{1-x^2}}dx$
• (b) $\int 2\sin \theta -3{\sec }^2\theta d\theta$
• (c) $\int 2\sec (3x)\tan (3x)-4\cos (x/5)dx$
• (d) $\int 2^x+e^{2x}dx$

02

Computing a definite integral from geometry

Find the value of this definite integral by sketching the integrand and considering areas:

$${\int }_0^{9}x-3dx$$

03

Integration - algebra rules

Suppose we know these three facts:

$${\int }_1^{4}f(x)dx=7,{\int }_2^{4}f(x)dx=5,{\int }_1^{4}g(x)dx=2.$$

• (a) Compute ${\int }_1^{4}4f(x)dx$
• (b) Compute ${\int }_1^{4}g(x)-f(x)dx$
• (c) Compute ${\int }_1^{4}8f(x)-7g(x)dx$
• (d) Compute ${\int }_1^2-f(x)dx$
• (e) Compute ${\int }_4^{1}g(x)dx$

04

Integral approximation methods

Soon we will learn that ${\int }_1^2\frac{dx}{x}={\ln (x)|}_1^2=\ln 2-\text{cancelto}0\ln 1=\ln (2)$.

Compute $L_2$ and $R_2$ and $M_2$ for three approximations of $\ln 2$ using this integral calculation. Do not use a calculator!

Draw the rectangles or trapezoids involved in each approximation. Which approximation do you think is the most accurate based on our drawings?

(Recall $L_2$ is the left endpoint rule, $R_2$ is the right endpoint rule, and $M_2$ is the midpoint rule.)