\(\int\left(x^2+1\right)e^2xdx\)

\(=\int\left(e^2x^3+e^2x\right)dx\)

\(=\dfrac{e^2x^4}{4}+\dfrac{e^2x^2}{2}+C\)

1. Rebecca/ always/ love/ hike/ and /other outdoor activities

=> Rebecca always loves hiking and other outdoor activities.

2. Fruit/ be/ convenient source/ of vitamin/ and energy

=> Fruits are a convenient source of vitamin and energy.

3. She/ not/ like/ fish/ because/ she/ hate/ thebones

=> She doesnt like fish because she hates the bones.

5. Watch/ TV/ too much/ not/ be/ good/ your eyes

=> Watching TV too much is not good for your eyes.

6. Many towns/ be/ full of/ shopping malls/ and/ fast/ food/ restaurants

=> Many towns are full of shopping malls and fast-food restaurants. 

7. We/ should/ keep/ healthy/ by/ eat/ well and/ exercise/ regularly

=> We should keep healthy by eating well and exercising regularly.

8. Nutrition and exercise/ be/ essential/ fitness and healthy

Khi chất điểm đi qua vị trí cân bằng thì tốc độ của nó \(v=v_{max}=\omega A=2.1=2\)(m/s)

\(\dfrac{\left(13\dfrac{1}{4}-2\dfrac{5}{27}-10\dfrac{5}{6}\right).230\dfrac{1}{25}+46\dfrac{3}{4}}{\left(1\dfrac{3}{7}+\dfrac{10}{3}\right):\left(12\dfrac{1}{3}-14\dfrac{2}{7}\right)}\)

\(=\dfrac{\left(\dfrac{53}{4}-\dfrac{59}{27}-\dfrac{65}{6}\right).\dfrac{5751}{25}+\dfrac{187}{4}}{\left(\dfrac{10}{7}+\dfrac{10}{3}\right):\left(\dfrac{37}{3}-\dfrac{100}{7}\right)}\)

\(=\dfrac{\left(\dfrac{1431}{108}-\dfrac{236}{108}-\dfrac{1170}{108}\right).\dfrac{5751}{25}+\dfrac{187}{4}}{\left(\dfrac{30}{21}+\dfrac{70}{21}\right):\left(\dfrac{259}{21}-\dfrac{300}{21}\right)}\)

\(=\dfrac{\dfrac{25}{108}.\dfrac{5751}{25}+\dfrac{187}{4}}{\dfrac{100}{21}:-\dfrac{41}{21}}\)

\(=\dfrac{\dfrac{213}{4}+\dfrac{187}{4}}{\dfrac{100}{21}\cdot-\dfrac{21}{41}}=\dfrac{100}{-\dfrac{100}{41}}=-41\)

1. They used to get up early in the morning in 2022.

2. They have just drawn pictures of their school.

3. He and she were workers in the factory in 2018.

1.people find out that children are more susceptible to depression when they have to face family violence.

=> It is found out that children are more susceptible to depression when they have to face family violence.

=> Children are found out to be more susceptible to depression when they have to face family violence.

2.some people believe that all the hard work will pay off.

=> It is believed that all the hard work will pay off.

=> All the hard work is believed to pay off.

3.they suppose that she works as a secretary in this company.

=> It is supposed that she works as a secretary in this company.

=> She is supposed to work as a secretary in this company.

4. some people believe that Santa Claus is the man who brings Christmas gifts to people.

=> It is believed that Santa Claus is the man who brings Christmas gifts to people.

=> Santa Claus is believed to be the man who brings Christmas gifts to people.

5, they acknowledged that they made up this story. 

=> It was acknowledged that they made up this story. 

=> They were acknowledged to have made up this story. 

6. someone reported that the amount of water consumed last month increased.

=> It was reported that the amount of water consumed last month increased.

=> The amount of water consumed last month was reported to increase.

7.someone believes that teenagers cannot live without the internet.

=> It is believed that teenagers cannot live without the internet.

=> Teenagers is believed not to be able to live without the internet.

8. people suppose that violence must be banned.

=> It is supposed that violence must be banned.

=> Violence is supposed to be banned.

9. they think that disasters wil become more and more terrible in the future.

=> It is thought that disasters wil become more and more terrible in the future.

=> Disasters are thought to become more and more terrible in the future.

\(cos\left(a+b\right)cos\left(a-b\right)\)

\(=\dfrac{1}{2}\left[cos\left(a+b+a-b\right)+cos\left(a+b-a+b\right)\right]\)

\(=\dfrac{1}{2}\left(cos2a+cos2b\right)\)

\(=\dfrac{1}{2}\left(2cos^2a-1+2cos^2b-1\right)\)

\(=\dfrac{1}{2}\left(2.\left(\dfrac{3}{5}\right)^2-1+2.\left(-\dfrac{4}{5}\right)^2-1\right)=0\)

Vì AB//CD nên B+C=180 (2 góc trong cùng phía)

Mà B=C => B=C=90 => BC vuông góc CD