Thread: How to solve integral of high degree polynomial without expansion?

1. How to solve integral of high degree polynomial without expansion?

How to solve integral of polynomial for example $$\int {(x^2 - 3x)}^5$$
Can we solve this without expansion $$\int {(2x - 6)}^3 dx$$

2. Re: How to solve integral of high degree polynomial without expansion?

the second one we can do using substitution

$u = 2x-6$

$du = 2 ~dx$

$\displaystyle \int (2x-6)^3~dx \Rightarrow \int u^3~\dfrac{du}{2} = \dfrac{u^4}{8} = \dfrac{(2x-6)^4}{8} = 2(x-3)^4$

One way or another you're going to have to expand the first one.

3. Re: How to solve integral of high degree polynomial without expansion?

I see.. thanks. The answer is $$2x^4 - 24x^3 + 108x^2 - 216x + C$$
And the expansion of $$(2x - 6)^4/8$$ is not like the answer.

4. Re: How to solve integral of high degree polynomial without expansion?

For the second one, I would start by factoring out $2^3$: $8 \int (x- 3)^3 dx$. Let u= x- 3 so du= dx. The integral becomes $8 \int u^3 du= \frac{8}{4} u^4+ C= 2 (x- 3)^4+ C$.

You say "the expansion of $(2x- 6)^4/8$ is not like the answer."

The expansion is $2(x^4- 12x^3+ 54x^2- 108x+ 81)= 2x^4- 24x^3+ 108x^3- 216x+ 162$. What is "the answer"?

Ok thank you

6. Re: How to solve integral of high degree polynomial without expansion?

Originally Posted by Helly123
How to solve integral of polynomial for example $$\int {(x^2 - 3x)}^5$$
None answered the first one. If you know that: $(x^3-3x)^5=x^{15}-15x^{13}+90x^{11}-270x^9+405x^7-243x^5$
wouldn't it be easy to find $\int {{{({x^3} - 3x)}^5}}~?$ See here.

7. Re: How to solve integral of high degree polynomial without expansion?

Originally Posted by Plato
None answered the first one. If you know that: $(x^3-3x)^5=x^{15}-15x^{13}+90x^{11}-270x^9+405x^7-243x^5$
wouldn't it be easy to find $\int {{{({x^3} - 3x)}^5}}~?$ See here.
the question did ask how to do it without expanding it

8. Re: How to solve integral of high degree polynomial without expansion?

Originally Posted by Plato
None answered the first one. If you know that: $(x^3-3x)^5=$
The integrand is $(x^2 - 3x)^5$.

9. Re: How to solve integral of high degree polynomial without expansion?

Originally Posted by romsek
the question did ask how to do it without expanding it
That is my whole point. Why not just expand it. Doing so really does simplify the task.
With the tools available now a lot of topics that we thought were important are now obsolete. Partial fractions for example.