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
tensa zangetsu [6.8K]
3 years ago
7

Calculus 2 Master needed, evaluate the indefinite integral of:

%20%5C%2C%20dx" id="TexFormula1" title="\int\( (lnx)^2} \, dx" alt="\int\( (lnx)^2} \, dx" align="absmiddle" class="latex-formula"> So far I had applied the integration by parts and got: (ln x )^2*x - \int\ x*((lnx)^2/2)} do we do integration by parts again? also, do we simplify the x at my current stage? steps shown would be appreciated
Mathematics
1 answer:
viva [34]3 years ago
8 0

Answer:

\int (\ln(x))^2dx=x(\ln(x)^2-2\ln(x)+2)+C

Step-by-step explanation:

So we have the indefinite integral:

\int (\ln(x))^2dx

This is the same thing as:

=\int 1\cdot (\ln(x))^2dx

So, let's do integration by parts.

Let u be (ln(x))². And let dv be (1)dx. Therefore:

u=(\ln(x))^2\\\text{Find du. Use the chain rule.}\\\frac{du}{dx}=2(\ln(x))\cdot\frac{1}{x}

Simplify:

du=\frac{2\ln(x)}{x}dx

And:

dv=(1)dx\\v=x

Therefore:

\int (\ln(x))^2dx=x\ln(x)^2-\int(x)(\frac{2\ln(x)}{x})dx

The x cancel:

=x\ln(x)^2-\int2\ln(x)dx

Move the 2 to the front:

=x\ln(x)^2-2\int\ln(x)dx

(I'm not exactly sure how you got what you got. Perhaps you differentiated incorrectly?)

Now, let's use integrations by parts again for the integral. Similarly, let's put a 1 in front:

=x\ln(x)^2-2\int 1\cdot\ln(x)dx

Let u be ln(x) and let dv be (1)dx. Thus:

u=\ln(x)\\du=\frac{1}{x}dx

And:

dv=(1)dx\\v=x

So:

=x\ln(x)^2-2(x\ln(x)-\int (x)\frac{1}{x}dx)

Simplify the integral:

=x\ln(x)^2-2(x\ln(x)-\int (1)dx)

Evaluate:

=x\ln(x)^2-2(x\ln(x)-x)

Now, we just have to simplify :)

Distribute the -2:

=x\ln(x)^2-2x\ln(x)+2x

And if preferred, we can factor out a x:

=x(\ln(x)^2-2\ln(x)+2)

And, of course, don't forget about the constant of integration!

=x(\ln(x)^2-2\ln(x)+2)+C

And we are done :)

You might be interested in
53.01 as a mixed number
elixir [45]
53 1/10 is the mixed number
5 0
3 years ago
Read 2 more answers
Calculate average rate of change between n=1 and n= 11
crimeas [40]

Answer:

The ordered pairs for 1 and 3 are:

(1,2) (3,8)

Now use the slope formula:

(y2-y1)/(x2-x1)

(8-2)/(3-1)= 6/2 = 3

Step-by-step explanation:

5 0
3 years ago
Read 2 more answers
<img src="https://tex.z-dn.net/?f=%5Csqrt%7B%20x%5E%7B2%7D-10x%2B25%7D%2B25%2B12%5Csqrt%7Bx%7D%20%3D15%5Csqrt%7Bx%7D" id="TexFor
vodomira [7]

Step-by-step explanation:

\sqrt{ x^{2}-10x+25}+25+12\sqrt{x} =15\sqrt{x} \\  \\  \therefore \: \sqrt{ x^{2}-10x+ {5}^{2} }+25 =15\sqrt{x} - 12\sqrt{x}\\  \\  \therefore \: \sqrt{( x - 5)^{2}}+25 =3\sqrt{x}  \\  \\ \therefore \:  x - 5+ 25 = 3 \sqrt{x}  \\  \\ \therefore \: x + 20 = 3 \sqrt{x}  \\  \\ squaring \: both \: sides \\ (x + 20)^{2}  = ( {3 \sqrt{x} })^{2}  \\ \therefore \:  {x}^{2}  + 40x + 400 = 9x \\ \therefore \:  {x}^{2}  + 40x + 400  - 9x = 0 \\  \therefore \:  {x}^{2}  + 31x + 400  = 0 \\

8 0
3 years ago
The United States Air Force has 19 woman for every 81 men enlisted. If a squadron has 250 members, approximately how many are me
diamong [38]
About 203 men. Hope this helps!!!!
5 0
3 years ago
Read 2 more answers
Question 7(Multiple Choice Worth 1 points)
Oksi-84 [34.3K]

Answer:

i would say that i would take 5 days

Step-by-step explanation:

4 0
3 years ago
Other questions:
  • WILL MARK BRAINLIEST!!!!!
    14·1 answer
  • What is the rule?<br> (0,0),(1,4),(-1,-4)
    11·1 answer
  • How do I find Three noncollinear points
    9·1 answer
  • Which one of the following groups of numbers includes all prime numbers? A. 7, 17, 63, 67 B. 3, 11, 23, 31 C. 1, 3, 11, 23 D. 2,
    12·2 answers
  • Lawrence buys his friends candy in the morning for $3 a box, but forgot some of his friends, so he went back to the candy shop a
    11·1 answer
  • What are all of the factors of 24 and 27
    5·1 answer
  • PLEASE HELP I WILL GIVE BRAINLIEST
    12·1 answer
  • Hey will give brainliest i will be very thankful thx!!!!!
    15·1 answer
  • Please help me :(bbbbbb
    10·1 answer
  • A sun's rays are vertical at the point A on a planet. At the point B, which is 2105 miles due north of A, the angle of the sun m
    11·2 answers
Add answer
Login
Not registered? Fast signup
Signup
Login Signup
Ask question!