Mastering Integration Techniques: Solving Intx2(1-5x2)10dx with Expert Help

[Dell]

need urgent help with this

any ideas will be appreciated

$$\int$$x²(1-5x²)¹⁰dx

tried all kinds of things, what i now think needs to be done is to change the differential to d(1-5x²) so that i won't have to open the (1-5x²)¹⁰ which i really don't want to nor do i think i am expected to.

d(1-5x²)=-10xdx... not really helping me :(

 

[Mark44]

I tried several approaches, among them substitution and integration by parts, but didn't seem to get anywhere with either of these.

Another approach that seems promising is a trig substitution, using x = 1/sqrt(5) * cos(theta).

Using this approach I was able to rewrite the integral as
$$-1/25 \int cos^2(\theta) sin^{21}(\theta)d\theta$$
This could be rewritten as two integrals, both in powers of sin(theta). At that point, I would use a table of integrals.

This approach might be beyond your present capabilities, but it's the only one I can think of that leads anywhere.

 

[Unco]

any ideas will be appreciated

$$\int$$x²(1-5x²)¹⁰dx

tried all kinds of things, what i now think needs to be done is to change the differential to d(1-5x²) so that i won't have to open the (1-5x²)¹⁰ which i really don't want to nor do i think i am expected to.

d(1-5x²)=-10xdx... not really helping me :(

You could just expand the integrand using the binomial theorem.