$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

# 4.0E: Exercises

$$\newcommand{\vecs}[1]{\overset { \scriptstyle \rightharpoonup} {\mathbf{#1}} }$$ $$\newcommand{\vecd}[1]{\overset{-\!-\!\rightharpoonup}{\vphantom{a}\smash {#1}}}$$$$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\id}{\mathrm{id}}$$ $$\newcommand{\Span}{\mathrm{span}}$$ $$\newcommand{\kernel}{\mathrm{null}\,}$$ $$\newcommand{\range}{\mathrm{range}\,}$$ $$\newcommand{\RealPart}{\mathrm{Re}}$$ $$\newcommand{\ImaginaryPart}{\mathrm{Im}}$$ $$\newcommand{\Argument}{\mathrm{Arg}}$$ $$\newcommand{\norm}[1]{\| #1 \|}$$ $$\newcommand{\inner}[2]{\langle #1, #2 \rangle}$$ $$\newcommand{\Span}{\mathrm{span}}$$

## Exercise $$\PageIndex{1}$$

1. Define the term “antiderivative” in your own words

2. Is it more accurate to refer to “the” antiderivative of $$f(x)$$ or “an” antiderivative of $$f(x)$$?

3. Use your own words to define the indefinite integral of $$f(x)$$.

4. Fill in the blanks: “Inverse operations do the ____ things in the _____ order.”

5. What is an “initial value problem”?

6. The derivative of a position function is a _____ function.

7. The antiderivative of an acceleration function is a ______ function.

Under Construction

## Exercise $$\PageIndex{2}$$

Evaluate the indefinite integrals:

1. $$\int 3x^3 \,dx$$

2. $$\int x^8 \,dx$$

3. $$\int (10x^2-2) \,dx$$

4. $$\int \,dt$$

5. $$\int 1 \,ds$$

6. $$\int \frac{1}{3t^2}\, dt$$

7. $$\int \frac{1}{t^2}\, dt$$

8. $$\int \frac{1}{\sqrt{x}}\, dx$$

9. $$\int \sec^2 \theta\, d\theta$$

10. $$\int \sin \theta\, d\theta$$

11. $$\int (\sec x \tan x +\csc x \cot x )\, dx$$

12. $$\int 5e^\theta\, d\theta$$

13. $$\int 3^t\, dt$$

14. $$\int \frac{5^t}{2}\, dt$$

15. $$\int (2t+3)^2\, dt$$

16. $$\int (t^2+3)(t^3-2t)\, dt$$

17. $$\int x^2x^3\, dx$$

18. $$\int e^\pi\, dx$$

19. $$\int a\, dx$$

Under Construction

## Exercise $$\PageIndex{3}$$

This problem investigates why Theorem 35 states that $$\int \frac{1}{x}\,dx = \ln |x|+C$$.
(a) What is the domain of $$y=\ln x$$?
(b) Find $$\frac{d}{dx}(\ln x)$$.
(c) What is the domain of $$y=\ln (-x)$$?
(d) Find $$\frac{d}{dx}\left ( (\ln (-x)\right )$$.
(e) You should find that $$1/x$$ has two types of antiderivatives, depending on whether $$x>0$$ or $$x<0$$. In one expression, give a formula for $$\int \frac{1}{x}\,dx$$ that takes these different domains into account, and explain your answer.

Under Construction

## Exercise $$\PageIndex{4}$$

Find $$f(x)$$ described by the given initial value problem.

1. $$f'(x)=\sin x\text{ and }f(0)=2$$

2. $$f'(x)=5e^x\text{ and }f(0)=10$$

3. $$f'(x)=4x^3-3x^2\text{ and }f(-1)=9$$

4. $$f'(x)=\sec^2 x\text{ and }f(\pi/4)=5$$

5. $$f'(x)=7^x\text{ and }f(2)=1$$

6. $$f''(x)=5\text{ and }f'(0)=7,f(0)=3$$

7. $$f''(x)=7x\text{ and }f'(1)=-1,f(1)=10$$

8. $$f''(x)=5e^x\text{ and }f'(0)=3,f(0)=5$$

9. $$f''(\theta)=\sin \theta \text{ and }f'(\pi)=2,f(\pi)=4$$

10. $$f''(x)=24x^2+2^x-\cos x \text{ and }f'(0)=5,f(0)=0$$

11. $$f''(x)=0\text{ and }f'(1)=3,f(1)=1$$

Under Construction

## Exercise $$\PageIndex{5}$$

Use information gained from the first and second derivative to sketch $$f(x)=\frac{1}{e^x+1}$$.

## Exercise $$\PageIndex{6}$$
Given $$y=x^2e^x\cos x$$, find $$dy$$.