Antiderivative by substitution of (2x^4)/(3x^5 + 4)

[Jade]

Antiderivatives are new to me. Some seem easy to me such as:

. . .the antiderivative of x^5 is 1/6x^6

. . .the antiderivative of (3x+1) is 3/2x^2 + x

But then they get a little more difficult and I get frustrated For instance:

. . .(2x^4)/(3x^5 + 4)

My answer is (2/5x^5)/(3/6x^6 + 4x)

then (1)/(x(ln x)^2)

I know the derivative of ln x is 1/x, so the antidervative would be that backwards. I know that this would be solved by substitution, but that is what I am having trouble with. :?
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Edited by stapel -- Reason for edit: clarity

 

[galactus]

Re: Antiderivative by substitution

but then they get a little more difficult and I get frustrated
(2x^4)/(3x^5 + 4) my answer is (2/5x^5)/(3/6x^6 + 4x)

Like anything else. it's a matter of practice and learning to see what works.

\(\displaystyle \L\\\int\frac{2x^{4}}{3x^{5}+4}dx\)


If you let \(\displaystyle \L\\u=3x^{5}+4\), then \(\displaystyle \L\\du=15x^{4}dx\)

\(\displaystyle \L\\\frac{du}{15}=x^{4}dx\)

If you multiply both sides by 2:

\(\displaystyle \L\\\frac{2}{15}du=2x^{4}dx\)

See what the right side is?. Your numerator.

So, you have:

\(\displaystyle \L\\\frac{2}{15}\int\frac{1}{u}du\)

Now, it's easy. Ain't it?.

then (1)/(x(ln x)^2) I know the derivative of ln x is 1/x so the antidervative would be that backwards. I know that this would be solved by substitution, but that is what i am having trouble with

This one uses u-subbing also.

\(\displaystyle \L\\\int\frac{1}{x(ln(x))^{2}}dx\)

If you let \(\displaystyle \L\\u=ln(x)\), then \(\displaystyle \L\\du=\frac{1}{x}dx\)

Then you have:

\(\displaystyle \L\\\int\frac{1}{u^{2}}du\)

 

[Jade]

Not easy

I like plain old derivatives better - I need more practice on the substitution rule.

 

[tkhunny]

Re: Antiderivative by substitution

x^5 is 1/6x^6

It will also help if you work on your notation. This is very bad. Do you know why? I can think of two reasons.

 

[Jade]

For people to understand me better??

I do not know why? Should I use parentheses. :?

 

[stapel]

I do not know why? Should I use parentheses. :?

That might be why tutors mentions using them so often....

What you have posted, "1/6x^6", could mean any of the following:

. . . . .[1/(6x)]^6

. . . . .[(1/6)x]^6

. . . . .1 / (6x)^6

. . . . .1 / (6 x^6)

. . . . .(1/6) x^6

Which, if any, did you mean?

Eliz.

 

[Jade]

I see your point

I believe I meant [(1/6)x^(6)]

 

[tkhunny]

That's better.

You also stated "x^5 is 1/6x^6". That's just wrong. When you try to invent new shorthand, it rarely appears useful to others, or to you a day later. In context, this statement was a little more clear, but still unsatisfactory in my view. Others may disagree,

Write clearly and completely.

 

[galactus]

Better yet, if you plan on using the forum on a regular basis, take the time to learn a little LaTex.