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
Leno4ka [110]
3 years ago
9

How do I find the integral ∫10(x−1)(x2+9)dxint10/((x-1)(x^2+9))dx ?

Mathematics
1 answer:
defon3 years ago
4 0
\int\frac{10}{(x-1)(x^2+9)}\ dx=(*)\\\\\frac{10}{(x-1)(x^2+9)}=\frac{A}{x-1}+\frac{Bx+C}{x^2+9}=\frac{A(x^2+9)+(Bx+C)(x-1)}{(x-1)(x^2+9)}\\\\=\frac{Ax^2+9A+Bx^2-Bx+Cx-C}{(x-1)(x^2+9)}=\frac{(A+B)x^2+(-B+C)x+(9A-C)}{(x-1)(x^2+9)}\\\Updownarrow\\10=(A+B)x^2+(-B+C)x+(9A-C)\\\Updownarrow\\A+B=0\ and\ -B+C=0\ and\ 9A-C=10\\A=-B\ and\ C=B\to9(-B)-B=10\to-10B=10\to B=-1\\A=-(-1)=1\ and\ C=-1

(*)=\int\left(\frac{1}{x-1}+\frac{-x-1}{x^2+9}\right)\ dx=\int\left(\frac{1}{x-1}-\frac{x+1}{x^2+9}\right)\ dx\\\\=\int\frac{1}{x-1}\ dx-\int\frac{x+1}{x^2+9}\ dx=\int\frac{1}{x-1}-\int\frac{x}{x^2+9}\ dx-\int\frac{1}{x^2+9}\ dx=(**)\\\\\#1\ \int\frac{1}{x-1}\ dx\Rightarrow\left|\begin{array}{ccc}x-1=t\\dx=dt\end{array}\right|\Rightarrow\int\frac{1}{t}\ dt=lnt+C_1=ln(x-1)+C_1

\#2\ \int\frac{x}{x^2+9}\ dx\Rightarrow  \left|\begin{array}{ccc}x^2+9=u\\2x\ dx=du\\x\ dx=\frac{1}{2}\ du\end{array}\right|\Rightarrow\int\left(\frac{1}{2}\cdot\frac{1}{u}\right)\ du=\frac{1}{2}\int\frac{1}{u}\ du\\\\\\=\frac{1}{2}ln(u)+C_2=\frac{1}{2}ln(x^2+9)+C_2

\#3\ \int\frac{1}{x^2+9}\ dx=\int\frac{1}{x^2+3^2}\ dx=\frac{1}{3}tan^{-1}\left(\frac{x}{3}\right)+C_3\\\\therefore:\\\\\#1;\ \#2;\ \#3\Rightarrow(**)=ln(x-1)+C_1-\frac{1}{2}ln(x^2+9)+C_2-\frac{1}{3}tan^{-1}\left(\frac{x}{3}\right)+C_3

\boxed{=ln(x-1)-\frac{1}{2}ln(x^2+9)-\frac{1}{3}tan^{-1}\left(\frac{x}{3}\right)+C}



You might be interested in
What is [4-{8x÷4x}]×8​
Molodets [167]

Answer:

The answer you should be getting is : 16

Step-by-step explanation:

=\frac{64x}{4x}

=\frac{64}{4}

=16

Hope this helped you!

~YEET !!!

8 0
3 years ago
What is <br> 935 × 145 × 0 =
olchik [2.2K]

Answer: 0

Step-by-step explanation:

Ligit 0 cuz you are multimplying by 0

6 0
3 years ago
Read 2 more answers
X=18+9/2y help <br> Me please
Nana76 [90]

Answer:

x = 18 + 9y/2

x-18 = 9y/2

(x-18)x2=9y

2(x-18)=9y

2(x-18) / 9 = y

y= 2(x-18)/9

Step-by-step explanation:

<em>Simplify 9/2y to 9y/2</em>

<em>Then Subtract 18 from both sides</em>

<em>Now Multiply both sides by 2</em>

<em>Then Regroup the terms</em>

<em>now divide both sides by 9</em>

<em>Then you will Switch sides</em>

4 0
3 years ago
All of the expressions that have the same value as LaTeX: 892\div8892 ÷ 8. Group of answer choices LaTeX: 8920\div808920 ÷ 80 La
algol13

Answer:

(a) 8920\div80

(e) 0.892\div0.008

Step-by-step explanation:

Given

892\div8

Required

Select equivalent expression

8920\div80

894\div10

89.2\div0.08

8.92\div0.8

0.892\div0.008

Start  by solving the given expression (using a calculator)

892\div8 = 111.5

Then we solve the given options (using a calculator)

(a) 8920\div80

8920\div80 = 111.5

(b) 894\div10

894\div10 = 89.4

(c) 89.2\div0.08

89.2\div0.08 = 1115

(d) 8.92\div0.8

8.92\div0.8 = 11.15

(e) 0.892\div0.008

0.892\div0.008 = 111.5

The equivalent expressions are (a) and (e)

5 0
2 years ago
how much time would it take for an airplane to reach its destination if it traveled at an average speed of 790 kilometers/hour f
Finger [1]
<span>the answer issssssssssss,  3713000</span>
7 0
3 years ago
Other questions:
  • Kyle and Elijah are planning a road trip to California. Their car travels 3/4 of a mile per min If they did not stop driving, ho
    8·2 answers
  • 4-3x &gt; 2x -1 what does it need to be to be greater or less than I guess
    8·1 answer
  • X=-2-8<br> 3x-4y=6<br> What is the answer for
    11·1 answer
  • Cydney.bulit a display stand out of two cubes. The large Cube is 8 inches on each side. The smaller cube is 4 inches on each sid
    7·1 answer
  • Umm whats y2 when y is -9 and whats z2 when z is 0.7 im not that smart so ill be asking alot
    6·1 answer
  • Please help ASAP!!!!!!
    14·1 answer
  • 25 POINTS!!!<br><br> x =<br><br> 4<br> 6<br> 12
    5·2 answers
  • How to convert 7,293 a into a fraction
    11·2 answers
  • Which statement best predicts the weather for Denver, Colorado on day 3.
    13·2 answers
  • Can you guys please help me on my homework? It would mean a lot to me :)
    13·1 answer
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!