1answer.
Ask question
Login Signup
Ask question
All categories
  • English
  • Mathematics
  • Social Studies
  • Business
  • History
  • Health
  • Geography
  • Biology
  • Physics
  • Chemistry
  • Computers and Technology
  • Arts
  • World Languages
  • Spanish
  • French
  • German
  • Advanced Placement (AP)
  • SAT
  • Medicine
  • Law
  • Engineering
ankoles [38]
2 years ago
12

Find the indefinite integral. (Note: Solve by the simplest method—not all require integration by parts. Use C for the constant o

f integration.)
\int 9arctan x dx

\int 9 arctan x dx
Mathematics
2 answers:
Lapatulllka [165]2 years ago
6 0

Answer:

\int{9 \arctan{\left(x \right)} d x} = 9 x  \arctan{\left(x \right)} - \frac{9}{2} \ln{\left(\left|{x^{2} + 1}\right| \right)}+C

Step-by-step explanation:

To find \int \:9\arctan \left(x\right)dx you must:

Step 1: Apply the constant multiple rule \int c f{\left(x \right)}\, dx = c \int f{\left(x \right)}\, dx with c = 9 and f{\left(x \right)} = \arctan {\left(x \right)}

\int{9 \arctan }{\left(x \right)} d x}} =9 \int{\arctan {\left(x \right)} d x}

Step 2: For the integral \int{\arctan}{\left(x \right)} d x}, use integration by parts \int {u} {dv}                    ={u}{v} -                    \int {v}{du}

Let {u}={\arctan}{\left(x \right)} and dv=dx.

Then

{du}=\left({\arctan}{\left(x \right)}\right)^{\prime }dx=\frac{dx}{x^{2} + 1} and {v}=\int{1 d x}=x

The integral can be rewritten as

9 {\int{\arctan}{\left(x \right)} d x}}=9 {\left(\arctan{\left(x \right)} \cdot x-\int{x \cdot \frac{1}{x^{2} + 1} d x}\right)}=9{\left(x\arctan{\left(x \right)} - \int{\frac{x}{x^{2} + 1} d x}\right)}

Let u=x^{2} + 1

Then du=\left(x^{2} + 1\right)^{\prime }dx = 2 x dx and we have that x dx = \frac{du}{2}.

Therefore,

9 x \arctan{\left(x \right)} - 9 {\int{\frac{x}{x^{2} + 1} d x}} = 9 x \arctan{\left(x \right)} - 9 {\int{\frac{1}{2 u} d u}}

9 x \arctan{\left(x \right)} - 9 {\int{\frac{1}{2 u} d u}} = 9 x \arctan{\left(x \right)} - 9 {\left(\frac{1}{2} \int{\frac{1}{u} d u}\right)}

Step 3: The integral of \frac{1}{u} is \int{\frac{1}{u} d u} = \ln{\left(u \right)}

x \arctan{\left(x \right)} - \frac{9}{2} {\int{\frac{1}{u} d u}} = 9 x \arctan{\left(x \right)} - \frac{9}{2} {\ln{\left(u \right)}}

Step 4: Recall that u=x^{2} + 1

9 x \arctan{\left(x \right)} - \frac{9}{2} \ln{\left({u} \right)} = 9 x \arctan{\left(x \right)} - \frac{9}{2} \ln{\left({\left(x^{2} + 1\right)} \right)}

Therefore,

\int{9 \arctan{\left(x \right)} d x} = 9 x  \arctan{\left(x \right)} - \frac{9}{2} \ln{\left(\left|{x^{2} + 1}\right| \right)}

Step 5: Add the constant of integration

\int{9 \arctan{\left(x \right)} d x} = 9 x  \arctan{\left(x \right)} - \frac{9}{2} \ln{\left(\left|{x^{2} + 1}\right| \right)}+C

umka2103 [35]2 years ago
3 0

Answer:

∫▒〖arctan⁡(x).1 dx=arctan⁡(x).x〗-1/2  ln⁡(1+x^2 )+C

Step-by-step explanation:

∫▒〖1st .2nd dx=1st∫▒〖2nd dx〗-∫▒〖(derivative of 1st) dx∫▒〖2nd dx〗〗〗

Let 1st=arctan⁡(x)

And 2nd=1

∫▒〖arctan⁡(x).1 dx=arctan⁡(x) ∫▒〖1 dx〗-∫▒〖(derivative of arctan(x))dx∫▒〖1 dx〗〗〗

As we know that  

derivative of arctan(x)=1/(1+x^2 )

∫▒〖1 dx〗=x

So  

∫▒〖arctan⁡(x).1 dx=arctan⁡(x).x〗-∫▒〖(1/(1+x^2 ))dx.x〗…………Eq1

Let’s solve ∫▒(1/(1+x^2 ))dx by substitution now  

Let 1+x^2=u

du=2xdx

Multiply and divide ∫▒〖(1/(1+x^2 ))dx.x〗 by 2 we get

1/2 ∫▒〖(2/(1+x^2 ))dx.x〗=1/2 ∫▒(2xdx/u)  

1/2 ∫▒(2xdx/u) =1/2 ∫▒(du/u)  

1/2 ∫▒(2xdx/u) =1/2  ln⁡(u)+C

1/2 ∫▒(2xdx/u) =1/2  ln⁡(1+x^2 )+C

Putting values in Eq1 we get

∫▒〖arctan⁡(x).1 dx=arctan⁡(x).x〗-1/2  ln⁡(1+x^2 )+C  (required soultion)

You might be interested in
Pls help I’ll brainlest ASAP
Volgvan

Answer:

sx

Step-by-step explanation:

5 0
2 years ago
If d(x)=9x^2 and g(x)=5x+7 then d(x)-g(x) can be rewritten as
Angelina_Jolie [31]

Answer:

9x^2-5x-7

Step-by-step explanation:

d(x)=9x^2\\g(x)=5x+7\\d(x)-g(x)=\\(9x^2)-(5x+7)=\\9x^2-5x-7

Hope this helps!

7 0
3 years ago
Read 2 more answers
The longest human power sporting event is the tour de France cycling race in a particular year the average speed for the winner
sergey [27]

Answer:

The winner completed the race in 96 hours and 52 minutes.

Step-by-step explanation:

Given:

Distance of the cycling race = 2292 miles

Average speed of the winner = 23.66\  mph

We need to find time required by winner to complete the race.

Solution:

Now we know that;

Time required can be calculated by dividing Total Distance from the Average speed.

framing in equation form we get;

time required by winner to complete the race = \frac{2292}{23.66} = 96.87\ hrs

Now converting 0.87\ hrs into minutes we get;

0.87\times 60= 52.2\approx 52\ mins

Hence the winner completed the race in 96 hours and 52 minutes.

3 0
2 years ago
the rain began very slowly, then got much heavier and finally tapered off. which graph models the situation
Virty [35]
Where’s the model????
4 0
3 years ago
Read 2 more answers
What is an equation of the line that passes through the point (- 5, - 6) and is parallel to the line 4x - 5y = 35
Elena L [17]

Answer:

4x - 5y = 10

Step-by-step explanation:

Any line parallel to 4x - 5y = 35 will have the same equation EXCEPT that the constant will be different.

Starting with 4x - 5y = 35, replace x with the given x-coordinate -5 and the given y-coordinate -6, and finally the given 35 with the constant C:

4(-5) - 5(-6) = C, or

-20 + 30 = C.  Thus, C = 10, and the equation of the new line is

4x - 5y = 10

7 0
3 years ago
Other questions:
  • Yesterday, the movie theater brought in $1,440 in ticket sales and $295.25 in food sales. Today, ticket sales were three-fourths
    11·2 answers
  • There are 10 players on the soccer team. Three of the players are in fifth grade. Five of the players are in the sixth grade. Wh
    6·2 answers
  • Will mark brainliest. Please answer:
    12·1 answer
  • How many right angles are there in the diagram?<br> There are<br> right angles.
    13·1 answer
  • 1. Solve the following equation by factoring: 4x2 + 12x + 9 = 0
    12·1 answer
  • Consider the sequence 16,-8,4,-2,1,
    12·1 answer
  • BRAINLIEST TO THE FIRST PERSON TO ANSWER!!The expression on the left side of an equation is shown below.
    8·1 answer
  • Will mark brainliest for having the steps.<br> prove tan(2x) = tan(x+x)
    11·2 answers
  • 2/5× = ?<br><br><br>I don't understand how to answer this​
    6·1 answer
  • Hi I did 1st 2 but need help with rest simplifying these and I need them explained step by step plz and thank you
    7·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!