Question 2, i tried to do it but couldn't.

 

In your first attempt you are doing things which are not allowed.

\((x-x^3)^{\frac13}\quad \ne \quad x^{\frac13}-x\)

 

How can i solve further without simplifying?
Besides i only know this method to solve problems, simplify and apply the formula.

 

Your very first step shows that you have a fundamental weakness in your algebra skills. You start of by asserting that

\(
{\(x-x^3\)}^{\frac{1}{3}} \, = \, x^{\frac{1}{3}}-x
\)

By this line of reasoning,

\(
{\(a+b)}^{2} \, = \, a^2 + b^2
\)

Do you agree with this?

If you do, then just set a=b=1 and see if it actually works out.

 

How can i solve further without simplifying?
Besides i only know this method to solve problems, simplify and apply the formula.

What method?

 

No i don't agree with that (a+b)²= a²+2ab+b²

 

Then can you tell me or give me a link of how to solve such problems?

 

One route you can go is to just differentiate the possible answers and see which one yields the original integrand. I would do the one that has "+1+C" last since the 1 could be absorbed into the C which means that this answer is probably a distractor.

 

So that simplification will not work. You'll have to find another one.
As an aid to figuring this out did it occur to you to differentiate one or more of the answers?
Also if you can get the integral into the form

\(\int u\cdot du\)
then maybe you can solve it.

 

That would take a lot of time, differentiating all the options

 

So that simplification will not work. You'll have to find another one.
As an aid to figuring this out did it occur to you to differentiate one or more of the answers?

Like what?
Give me some hint or something iam not asking for complete solution

 

One route you can go is to just differentiate the possible answers and see which one yields the original integrand. I would do the one that has "+1+C" last since the 1 could be absorbed into the C which means that this answer is probably a distractor.

That would take a lot of time, differentiating all the options

 

So that simplification will not work. You'll have to find another one.
As an aid to figuring this out did it occur to you to differentiate one or more of the answers?
If you can get the integrand into the form

\(\int u\cdot du\)
then maybe you can solve it.

Differentiating all the options ia going to take alot of time, and iam going to get just over a minute to solve this kind of queation, i've got a test coming up in no time.

 

That would take a lot of time, differentiating all the options

Really? I did it in less than three minutes.

Look at the three options -- they all have the same primary component, namely

\(
{\( \frac{1}{x^2} - 1\)}^{\frac{4}{3}}
\)

How hard can it be?

 

Which is worse

1. Taking a bit of time
2. Passing on answering the question

 

Really? I did it in less than three minutes.

Look at the three options -- they all have the same primary component, namely

\(
{\( \frac{1}{x^2} - 1\)}^{\frac{4}{3}}
\)

How hard can it be?

It maybe easy for you.
But the answer is C, not A.

 

Which is worse

1. Taking a bit of time
2. Passing on answering the question

If i wanted to pass this question, i wouldn't have wasted half an hour tring to solve it.

 

OK, it's completely up to you how you approach your problems.

 

It maybe easy for you.
But the answer is C, not A.

Yes, the answer is C. Where did I, or anyone, say that the answer is A?

I said that all three of the possible answers could be evaluated by differentiating that one expression.

 

Let me walk you through it one step at a time.

What is

\(
\frac{d}{dx}\[{\( \frac{1}{x^2} - 1\)}^{\frac{4}{3}}\]
\)

I picked this form instead of the (1 - 1/x^2) form simply because this is the form used in two of the three potential answers. But hopefully you recognize that the differ only by a minus sign.