Integrals

Problem 1301

16. Find possible values of $a$ and $b$ that make the statement true. If possible, use a sketch to support your answer. (There may be more than one correct answer.) a. $\int_{-2}^{1} f(x) d x+\int_{1}^{5} f(x) d x=\int_{a}^{b} f(x) d x$ b. $\int_{-3}^{3} f(x) d x+\int_{3}^{6} f(x) d x-\int_{a}^{b} f(x) d x=\int_{-1}^{6} f(x) d x$ c. $\int_{a}^{b} \sin x d x<0$ d. $\int_{a}^{b} \cos x d x=0$
17. The graph of $f$ consists of line segments, as show in the figure below. Evaluate each definite integral by using geometric formulas. A. $\int_{0}^{1}-f(x) d x$ B. $\int_{11}^{0} f(x) d x$ C. $\int_{3}^{4} 6-f(x) d x$ D. $\int_{10}^{4} 3 f(x) d x$

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Problem 1302

10) The improper integral $\int_{0}^{\infty} \frac{16 \tan ^{-1} x}{1+x^{2}} d x$ A) diverges B) converges to $2 \pi^{2}$ C) converges to $\frac{1}{2} \pi^{2}$ D) converges to $\frac{3}{2} \pi^{2}$ E) NOTA

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Problem 1303

Unanswered
Question 11 Not yet graded / 1 pts
Evaluate the following improper integral if it is convergent. If it is not convergent, write divergent $\int_{0}^{\infty} e^{-2 x} d x$
Your Answer:

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Problem 1304

(d) $\int_{1}^{e} \frac{d}{d x}\left[\frac{x \ln x}{1+x^{2}}\right] d x$

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Problem 1305

3. A tea kettle is taken off of the stove and is cooling on the countertop for 10 minutes. $\mathrm{H}^{\prime}(t)$ , a differentiable function, represents the rate at which the temperature is changing, measured in degrees Celsius per minute, and $t$ is measured in minutes. \begin{tabular}{|c|c|c|c|c|c|} \hline $t(\mathrm{~min})$ & 0 & 2 & 5 & 9 & 10 \\ \hline $H^{\prime}(t)\left({ }^{\circ} \mathrm{C} / \mathrm{min}\right)$ & -2.1 & -1.8 & -1.6 & -1.2 & -0.8 \\ \hline \end{tabular} (c) If the temperature of the tea in the kettle was $96^{\circ} \mathrm{C}$ when it was taken off the stove, what is the temperature after 10 minutes?

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Problem 1306

Assume the annual rate of change in the national debt of a country (in billions of dollars per year) can be modeled by the function $D^{\prime}(t)=848.18+816.08 t-151.95 t^{2}+17.76 t^{3}$ where $t$ is the number of years since 1995. By how much did the debt increase between 1996 and $2007 ?$
The debt increased by $\$ 72,270.55$ billion. (Round to two decimal places as needed.)

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Problem 1307

Question 1 (a) Rationalize the denominator and simplify $\frac{2 \sqrt{3}-1}{1-\sqrt{3}}+\frac{3 \sqrt{3}}{1+\sqrt{3}}$ . [5 marks] (b) If $w=4+7 i$ , express $w+\frac{1}{w}$ in the form $a+b i$ where $a$ and $b$ are real. [5 marks]
Question 2 Given that $f(x)=\sqrt{x+5}$ and $g(x)=\ln (x+5)$ . (i) Sketch the graph of $f(x)$ . (ii) State the domain and range of $f(x)$ . (iii) Find $f^{-1}(x)$ and $\left(g \circ f^{-1}\right)(x)$ . [10 marks] Question 3 (a) Given that the $5^{\text {th }}$ term of an arithmetic progression is 21 and its $10^{\text {th }}$ term is 41 , find (i) the common difference, $d$ and the first term, $a$ . (ii) the sum of first $20^{\text {th }}$ term. [7 marks] (b) Expand $(2-3 x)^{8}$ in ascending power of $x$ up to the term in $x^{3}$ . [3 marks]
Question 4 (a) Given that $\mathbf{A}=\left(\begin{array}{cc}2 & -1 \\ 3 & 2\end{array}\right)$ and $\mathbf{B}=\left(\begin{array}{ll}a & 1 \\ b & 1\end{array}\right)$ where $a$ and $b$ are real. Find the values of $a$ and $b$ such that $\mathbf{A B}=\mathbf{B A}$ . [6 marks] (b) If $\mathbf{P}=\left(\begin{array}{ll}3 & -2 \\ 4 & -3\end{array}\right)$ , show that the inverse matrix of $\mathbf{P}$ is also $\mathbf{P}$ . [4 marks]
Question 5 (a) Given the parametric equations $x=t^{3}-t$ and $y=t^{2}+t$ where $t>0$ .
Find $\frac{d y}{d x}$ in terms of $t$ . [5 marks] (b) Evaluate $\int x \sqrt{x^{2}-5} d x$ by using the substitution $u=x^{2}-5$ . [5 marks]

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Problem 1308

Evaluate the integral. (Use C for the constant of integration.) $\int \frac{\ln (x)}{x^{2}} d x$

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Problem 1309

9) Знайти: $\int_{0}^{x} \frac{\operatorname{sh} t^{3}}{t^{3}} d t .$

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Problem 1310

point)
Evaluate $\int_{0}^{3} 2 x \sqrt{x^{2}+16} d x$
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Problem 1311

(1 point)
Use the Fundamental Theorem of Calculus to find $\int_{1}^{16} \frac{\sin (\sqrt[4]{x})}{\sqrt[4]{x^{3}}} d x=$ $\square$

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Problem 1312

Evaluate the definite integral. $\int_{-\pi / 2}^{\pi / 2} \sin ^{8}(x) \cos (x) d x=$ $\square$
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Problem 1313

Let $f(x)=\left\{\begin{array}{ll} 0 & \text { if } x<-3 \\ 2 & \text { if }-3 \leq x<-1 \\ -5 & \text { if }-1 \leq x<5 \\ 0 & \text { if } x \geq 5 \end{array}\right.$ and $g(x)=\int_{-3}^{x} f(t) d t$
Determine the value of each of the following: (a) $g(-5)=$ $\square$ (b) $g(-2)=$ $\square$ (c) $g(0)=$ $\square$ (d) $g(6)=$ $\square$ (e) The absolute maximum of $g(x)$ occurs when $x=$ $\square$ and is the value $\square$ It may be helpful to make a graph of $f(x)$ when answering these questions.
Note: You can earn partial credit on this problem.

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Problem 1314

oblem $1 \vee$ oblem $2 \checkmark$ oblem $3 \checkmark$ oblem 4 blem 5 blem $6 \checkmark$ blem $7 \checkmark$ blem 8 lem 9

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Problem 1315

(1 point) $\qquad$ Evaluate $\int_{\pi}^{\pi} \sin ^{4} x \cos ^{3} x d x$ Answer: $\square$
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Problem 1316

5.2 - The Definite Integral: Problem 2 (1 point)
The limit $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \sqrt{2 x_{i}^{}+\left(x_{i}^{}\right)^{2}} \Delta x$ can be expressed as a definite integral on the interval $[1,8]$ of the form $\int_{a}^{b} f(x) d x$
Determine $a, b$ , and $f(x)$ . $\begin{array}{l} a=\square \\ b=\square \end{array}$ $f(x)=$ $\square$

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Problem 1317

(1 point)
Consider the integral $\int_{2}^{6} \frac{x}{1+x^{5}} d x$ . Which of the following expressions represents the integral as a limit of Riemann sums? A. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2+\frac{4 i}{n}}{1+\left(2+\frac{4 i}{n}\right)^{5}}$ B. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{2+\frac{6 i}{n}}{1+\left(2+\frac{6 i}{n}\right)^{5}}$ C. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{6}{n} \frac{2+\frac{6 i}{n}}{1+\left(2+\frac{6 i}{n}\right)}$ D. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{4}{n} \frac{2+\frac{4 i}{n}}{1+\left(2+\frac{4 i}{n}\right)^{5}}$ E. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{4}{n} \frac{2+\frac{4 i}{n}}{1+\left(2+\frac{4 i}{n}\right)}$ F. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{6}{n} \frac{2+\frac{6 i}{n}}{1+\left(2+\frac{6 i}{n}\right)^{5}}$ Preview My Answers Submit Answers
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Problem 1318

Definition: The AREA A of the region $S$ that lies under the graph of the continuous function $f$ is the limit of the sum of the areas of approximating rectangles $A=\lim _{n \rightarrow \infty} R_{n}=\lim _{n \rightarrow \infty}\left[f\left(x_{1}\right) \Delta x+f\left(x_{2}\right) \Delta x+\ldots+f\left(x_{n}\right) \Delta x\right]$
Consider the function $f(x)=\frac{\ln (x)}{x}, 3 \leq x \leq 10$ . Using the above definition, determine which of the following expressions represents the area under the graph of $f$ as a limit. A. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{7}{n} \frac{\ln \left(3+\frac{7 i}{n}\right)}{3+\frac{7 i}{n}}$ B. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{7}{n} \frac{\ln \left(\frac{7 i}{n}\right)}{\frac{7 i}{n}}$ C. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{10}{n} \frac{\ln \left(\frac{10 i}{n}\right)}{\frac{10 i}{n}}$ D. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{\ln \left(3+\frac{7 i}{n}\right)}{3+\frac{7 i}{n}}$ E. $\lim _{n \rightarrow \infty} \sum_{i=1}^{n} \frac{10}{n} \frac{\ln \left(3+\frac{10 i}{n}\right)}{3+\frac{10 i}{n}}$ Preview My Answers Submit Answers

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Problem 1319

Find the most general antiderivative of the function $g(x)=\sqrt[7]{x^{6}}+\sqrt[6]{x^{7}}$
Answer: $G(x)=$ $\square$ (i) Preview My Answers Submit Answers

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Problem 1320

(1 point)
Find the function with derivative $f^{\prime}(x)=e^{9 x}$ that passes through the point $P=(0,2 / 9)$ . $f(x)=$ $\square$ Preview My Answers Submit Answers
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Problem 1321

Let $f(x)=\frac{19}{\sqrt{1-x^{2}}}$ . Enter an antiderivative of $f(x)$ . $\square$

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Problem 1322

Find an antiderivative of $q(t)=(t+4)^{2}$ $Q(t)=$ $\square$ Preview My Answers Submit Answers

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Problem 1323

(1) $\int^{3} x^{2} d x$

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Problem 1324

Find the area of the region enclosed by the curves $y=2 \cos x$ and $y=2 \cos 2 x$ for $0 \leq x \leq \pi$ .

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Problem 1325

Given $\vec{F}=4 \hat{i}+5 \hat{j}-6 y \hat{k}$ . Find $\oint \vec{F} \cdot \overrightarrow{d l}$ going around the loop that starts from the point $(0,0,0)$ to the point $(0,0,4)$ then to the point $(0,1,4)$ then to the point $(0,1,0)$ and back to $(0,0,0)$ . a. 4 b. -4 c. 24 d. 0 e. -24

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Problem 1326

Given $\vec{F}=4 \hat{i}+5 \hat{j}-6 y \hat{k}$ . Find $\oint \vec{F} \cdot \overrightarrow{d l}$ going around the loop that starts from the point $(0,0,0)$ to the point $(0,0,4)$ then to the paint $(0,1,4)$ then to the point $(0,1,0)$ and back to $(0,0,0)$ . a. -4 b. -24 c. 0 d. 24 e. 4
Clear my choice

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Problem 1327

3). Evaluate by changing of the ordrof integration. (a) $\int_{0}^{2} \int_{1}^{e^{x}} d y d x$ (b) $\int_{0}^{1} \int_{x^{2}}^{2-x} x y d x d y$ (c) $\int_{0}^{4 a} \int_{x / 4 a}^{2 \sqrt{a x}} d y d x$ .

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Integrals

Problem 1301

Problem 1302

10) The improper integral ∫0∞16tan⁡−1x1+x2dx\int_{0}^{\infty} \frac{16 \tan ^{-1} x}{1+x^{2}} d x∫0∞​1+x216tan−1x​dx A) diverges B) converges to 2π22 \pi^{2}2π2 C) converges to 12π2\frac{1}{2} \pi^{2}21​π2 D) converges to 32π2\frac{3}{2} \pi^{2}23​π2 E) NOTA

Problem 1303

UnansweredQuestion 11 Not yet graded / 1 ptsEvaluate the following improper integral if it is convergent. If it is not convergent, write divergent ∫0∞e−2xdx\int_{0}^{\infty} e^{-2 x} d x∫0∞​e−2xdxYour Answer:

Problem 1304

(d) ∫1eddx[xln⁡x1+x2]dx\int_{1}^{e} \frac{d}{d x}\left[\frac{x \ln x}{1+x^{2}}\right] d x∫1e​dxd​[1+x2xlnx​]dx

Problem 1305

Problem 1306

Problem 1307

Problem 1308

Evaluate the integral. (Use C for the constant of integration.) ∫ln⁡(x)x2dx\int \frac{\ln (x)}{x^{2}} d x∫x2ln(x)​dx

Problem 1309

9) Знайти: ∫0xsh⁡t3t3dt.\int_{0}^{x} \frac{\operatorname{sh} t^{3}}{t^{3}} d t .∫0x​t3sht3​dt.

Problem 1310

point)Evaluate ∫032xx2+16dx\int_{0}^{3} 2 x \sqrt{x^{2}+16} d x∫03​2xx2+16​dxAnswer: □\square□ Preview My Answers Submit Answers

Problem 1311

(1 point)Use the Fundamental Theorem of Calculus to find ∫116sin⁡(x4)x34dx=\int_{1}^{16} \frac{\sin (\sqrt[4]{x})}{\sqrt[4]{x^{3}}} d x=∫116​4x3​sin(4x​)​dx= □\square□

Problem 1312

Evaluate the definite integral. ∫−π/2π/2sin⁡8(x)cos⁡(x)dx=\int_{-\pi / 2}^{\pi / 2} \sin ^{8}(x) \cos (x) d x=∫−π/2π/2​sin8(x)cos(x)dx= □\square□Preview My Answers Submit Answers

Problem 1313

Problem 1314

oblem 1∨1 \vee1∨ oblem 2✓2 \checkmark2✓ oblem 3✓3 \checkmark3✓ oblem 4 blem 5 blem 6✓6 \checkmark6✓ blem 7✓7 \checkmark7✓ blem 8 lem 9

Problem 1315

(1 point) \qquad Evaluate ∫ππsin⁡4xcos⁡3xdx\int_{\pi}^{\pi} \sin ^{4} x \cos ^{3} x d x∫ππ​sin4xcos3xdx Answer: □\square□Preview My Answers Submit AnswersYou have attempted this problem 0 times. You have unlimited attempts remaining.

Problem 1316

Problem 1317

Problem 1318

Problem 1319

Find the most general antiderivative of the function g(x)=x67+x76g(x)=\sqrt[7]{x^{6}}+\sqrt[6]{x^{7}}g(x)=7x6​+6x7​Answer: G(x)=G(x)=G(x)= □\square□ (i) Preview My Answers Submit Answers

Problem 1320

(1 point)Find the function with derivative f′(x)=e9xf^{\prime}(x)=e^{9 x}f′(x)=e9x that passes through the point P=(0,2/9)P=(0,2 / 9)P=(0,2/9). f(x)=f(x)=f(x)= □\square□ Preview My Answers Submit AnswersYou have attempted this problem 0 times. You have unlimited attempts remaining.

Problem 1321

Let f(x)=191−x2f(x)=\frac{19}{\sqrt{1-x^{2}}}f(x)=1−x2​19​. Enter an antiderivative of f(x)f(x)f(x). □\square□

Problem 1322

Find an antiderivative of q(t)=(t+4)2q(t)=(t+4)^{2}q(t)=(t+4)2 Q(t)=Q(t)=Q(t)= □\square□ Preview My Answers Submit Answers

Problem 1323

(1) ∫3x2dx\int^{3} x^{2} d x∫3x2dx

Problem 1324

Find the area of the region enclosed by the curves y=2cos⁡xy=2 \cos xy=2cosx and y=2cos⁡2xy=2 \cos 2 xy=2cos2x for 0≤x≤π0 \leq x \leq \pi0≤x≤π.

Problem 1325

Problem 1326

Problem 1327

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10) The improper integral $\int_{0}^{\infty} \frac{16 \tan ^{-1} x}{1+x^{2}} d x$ A) diverges B) converges to $2 \pi^{2}$ C) converges to $\frac{1}{2} \pi^{2}$ D) converges to $\frac{3}{2} \pi^{2}$ E) NOTA

Unanswered
Question 11 Not yet graded / 1 pts
Evaluate the following improper integral if it is convergent. If it is not convergent, write divergent $\int_{0}^{\infty} e^{-2 x} d x$
Your Answer:

(d) $\int_{1}^{e} \frac{d}{d x}\left[\frac{x \ln x}{1+x^{2}}\right] d x$

Evaluate the integral. (Use C for the constant of integration.) $\int \frac{\ln (x)}{x^{2}} d x$

9) Знайти: $\int_{0}^{x} \frac{\operatorname{sh} t^{3}}{t^{3}} d t .$

point)
Evaluate $\int_{0}^{3} 2 x \sqrt{x^{2}+16} d x$
Answer: $\square$ Preview My Answers Submit Answers

(1 point)
Use the Fundamental Theorem of Calculus to find $\int_{1}^{16} \frac{\sin (\sqrt[4]{x})}{\sqrt[4]{x^{3}}} d x=$ $\square$

Evaluate the definite integral. $\int_{-\pi / 2}^{\pi / 2} \sin ^{8}(x) \cos (x) d x=$ $\square$
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oblem $1 \vee$ oblem $2 \checkmark$ oblem $3 \checkmark$ oblem 4 blem 5 blem $6 \checkmark$ blem $7 \checkmark$ blem 8 lem 9

(1 point) $\qquad$ Evaluate $\int_{\pi}^{\pi} \sin ^{4} x \cos ^{3} x d x$ Answer: $\square$
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Find the most general antiderivative of the function $g(x)=\sqrt[7]{x^{6}}+\sqrt[6]{x^{7}}$
Answer: $G(x)=$ $\square$ (i) Preview My Answers Submit Answers

(1 point)
Find the function with derivative $f^{\prime}(x)=e^{9 x}$ that passes through the point $P=(0,2 / 9)$ . $f(x)=$ $\square$ Preview My Answers Submit Answers
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Let $f(x)=\frac{19}{\sqrt{1-x^{2}}}$ . Enter an antiderivative of $f(x)$ . $\square$

Find an antiderivative of $q(t)=(t+4)^{2}$ $Q(t)=$ $\square$ Preview My Answers Submit Answers

(1) $\int^{3} x^{2} d x$

Find the area of the region enclosed by the curves $y=2 \cos x$ and $y=2 \cos 2 x$ for $0 \leq x \leq \pi$ .