How Do You Correctly Solve the Integral of [(3-ln2x)^3]/2x?

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In summary, the integral of [(3-ln2x)^3]/2x is equal to -1/16 * (3-ln2x)^4 + C, with the correct use of the chain rule and fixing the mistake with the 4th power.
  • #1
dangish
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question: integral of [(3-ln2x)^3]/2x

my workings:

I let u = 3-ln2x
then du= -2/x dx
so -1/2du = 1/x dx

this leaves me with -(1/2)*integral of u^3/2 du

I take the bottom 2 out to get -(1/2)*(1/2) * integral of u^3 du

which is -1/4 * (u^4)/4

then I sub u into get

-1/4 *(3-ln2x)/4

Which is -1/16 * (3-ln2x)

On my test I only got 8/10 for this question and When I plug this example into Wolfram it gives some wacky answer.

Can someone tell me if I'm right? Thanks
 
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  • #2
dangish said:
question: integral of [(3-ln2x)^3]/2x

my workings:

I let u = 3-ln2x
then du= -2/x dx

No. du = -(1/x)dx. You need the chain rule there.

so -1/2du = 1/x dx

this leaves me with -(1/2)*integral of u^3/2 du

I take the bottom 2 out to get -(1/2)*(1/2) * integral of u^3 du

which is -1/4 * (u^4)/4

then I sub u into get

-1/4 *(3-ln2x)4/4

You left off the 4th power. Fix those two mistakes and it is correct.
 
  • #3
Oops.. the 4th power I did write on my test just missed it there..

and damn I can't believe I got that derivative wrong haha

Thanks
 

FAQ: How Do You Correctly Solve the Integral of [(3-ln2x)^3]/2x?

What is the first step in integrating [(3-ln2x)^3]/2x?

The first step in integrating this expression is to use the power rule to expand the numerator, resulting in (27 - 54ln2x + 36ln^2(2x) - 8ln^3(2x))/2x.

How do you handle the ln(2x) term in the numerator?

The ln(2x) term can be simplified using the logarithm property log(ab) = log(a) + log(b). This results in ln^2(2x) = 2ln(2x). Therefore, the expression can be rewritten as (27 - 54ln2x + 72ln(2x) - 8ln^3(2x))/2x.

What is the next step after simplifying the ln(2x) term?

After simplifying the ln(2x) term, the next step is to use the quotient rule to separate the fraction into two parts. This results in (27/2x) - (54ln2x)/2x + (72ln(2x))/2x - (8ln^3(2x))/2x.

How do you integrate the first three terms?

The first three terms can be integrated using the power rule, resulting in (27/2x) - (54/x) + (72/x^2) + C. Note that the constant C is added to account for the integration constant.

What is the final result after integrating the entire expression?

The final result is (27/2x) - (54/x) + (72/x^2) - (4/x^3) + C.

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