Indefinite integrals

Question:
$\displaystyle \int \sqrt[3] {3x+5}.dx $

So the equation will be $\displaystyle \int (3x+5)^\frac{1}{3}.dx $

$\displaystyle u=3x+5 \rightarrow \int u^\frac{1}{3} $
$\displaystyle du=3 $
After this step i dont know what to do , do i divide by 3 to produce 1?
Please answer in step .Thank you

Quote:

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Originally Posted by mj.alawami

Question:
$\displaystyle \int \sqrt[3] {3x+5}.dx $

So the equation will be $\displaystyle \int (3x+5)^\frac{1}{3}.dx $

$\displaystyle u=3x+5 \rightarrow \int u^\frac{1}{3} $
$\displaystyle du=3 $

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Here's your problem! du is NOT equal to 3, du= 3dx.
And from that it should be easy to see that dx= (1/3)du.

Quote:

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After this step i dont know what to do , do i divide by 3 to produce 1?
Please answer in step .Thank you

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I have no idea what you could mean by "do i divide by 3 to produce 1?". Divide what by 3? "Produce 1" where and how? The more precise you are in mathematics, the bettter. Practise being precise.

Quote:

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Originally Posted by mj.alawami

Question:
$\displaystyle \int \sqrt[3] {3x+5}.dx $

So the equation will be $\displaystyle \int (3x+5)^\frac{1}{3}.dx $

$\displaystyle u=3x+5 \rightarrow \int u^\frac{1}{3} $
$\displaystyle du=3 $
After this step i dont know what to do , do i divide by 3 to produce 1?
Please answer in step .Thank you

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$\displaystyle u=3x+5$ then

$\displaystyle du=3dx$

$\displaystyle \frac{1}{3}du=dx$.

So, we have

$\displaystyle \int(3x+5)^{1/3}dx=\int{u^{1/3}}\frac{1}{3}du$

Quote:

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Originally Posted by HallsofIvy

Here's your problem! du is NOT equal to 3, du= 3dx.
And from that it should be easy to see that dx= (1/3)du.


I have no idea what you could mean by "do i divide by 3 to produce 1?". Divide what by 3? "Produce 1" where and how? The more precise you are in mathematics, the bettter. Practise being precise.

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Answer:
$\displaystyle \int \frac{1}{4} (3x+5)^\frac{4}{3} +C $

Is my answer Correct ?

Quote:

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Originally Posted by mj.alawami

Question:
$\displaystyle \int \sqrt[3] {3x+5}.dx $

So the equation will be $\displaystyle \int (3x+5)^\frac{1}{3}.dx $

$\displaystyle u=3x+5 \rightarrow \int u^\frac{1}{3} $
$\displaystyle du=3 $
After this step i dont know what to do , do i divide by 3 to produce 1?
Please answer in step .Thank you

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It might help to write it out like this

$\displaystyle u = 3x + 5$ so $\displaystyle \frac{du}{dx} = 3$.


Now work on the integral

$\displaystyle \int{(3x + 5)^{\frac{1}{3}}\,dx} = \frac{1}{3}\int{(3x + 5)^{\frac{1}{3}} \cdot 3\,dx}$

$\displaystyle = \frac{1}{3}\int{u^{\frac{1}{3}}\,\frac{du}{dx}\,dx }$

$\displaystyle = \frac{1}{3}\int{u^{\frac{1}{3}}\,du}$.


Might be an extra step but easier to avoid making mistakes...

Quote:

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Originally Posted by Prove It

It might help to write it out like this

$\displaystyle u = 3x + 5$ so $\displaystyle \frac{du}{dx} = 3$.


Now work on the integral

$\displaystyle \int{(3x + 5)^{\frac{1}{3}}\,dx} = \frac{1}{3}\int{(3x + 5)^{\frac{1}{3}} \cdot 3\,dx}$

$\displaystyle = \frac{1}{3}\int{u^{\frac{1}{3}}\,\frac{du}{dx}\,dx }$

$\displaystyle = \frac{1}{3}\int{u^{\frac{1}{3}}\,du}$.




Might be an extra step but easier to avoid making mistakes...

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Answer:
$\displaystyle \frac{1}{2} (3x+5)^\frac{4}{3} +C $

Is my answer Correct ?

No.

$\displaystyle \frac{1}{3}\int{u^{\frac{1}{3}}\,du} = \frac{1}{3}\cdot\frac{3}{4}u^{\frac{4}{3}} + C$

$\displaystyle = \frac{1}{4}u^{\frac{4}{3}} + C$

$\displaystyle = \frac{1}{4}(3x + 5)^{\frac{4}{3}} + C$.

Quote:

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Originally Posted by mj.alawami

Answer:
$\displaystyle \int \frac{1}{4} (3x+5)^\frac{4}{3} +C $

Is my answer Correct ?

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What do you get if you differentiate that? Did you even try that?

substitution isn't needed in this question.
If you remember this:

if you have a derivative that is in the form: (ax+b)^n
then the integral will be: [(ax+b)^n+1] / a(n+1)

so just substitute your question into that rule.

Quote:

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Originally Posted by bhuang

substitution isn't needed in this question.
If you remember this:

if you have a derivative that is in the form: (ax+b)^n
then the integral will be: [(ax+b)^n+1] / a(n+1)

so just substitute your question into that rule.

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Don't try remembering lots of rules, it gets to be too much of a headache.

It's easier to remember a few basic rules and to use substitution for the rest.

Besides, the rule you speak of comes from substitution anyway.