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### Where am I going wrong integrating $\int x^2(x^3-6)^{34} dx$ by parts?

Integration by parts is hardly needed. We can use substitution! Let $u = (x^3 - 6)$. So $du = 3x^2\,dx$. Given $$\int x^2(x^3-6)^{34} dx = \frac 13 \int 3x^2(x^3-6)^{34} dx,$$ under substitution, the ...
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### Simpler proof that $y^3[d^2y/dx^2]$ is a constant if $y^2=ax^2+bx+c$?

Here is a proof that uses only the fact that $u = ax^2 + bx + c$ is quadratic and hence $u''' = 0$. We know that $y^2 = u$ and hence $2 y y' = u' , \, y^4 = u^2$. Differentiate the second identity ...
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### Simpler proof that $y^3[d^2y/dx^2]$ is a constant if $y^2=ax^2+bx+c$?

Using differentials, it is easy to see that $$2y dy = (2ax+ b) dx \tag{1}$$ from which we can write $$u = \frac{dy}{dx} = \frac{2ax+b}{2y}$$ Using again differentials and product rule, \begin{...
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Just as spaces are important in separating words, so they are in mathematical notation. Thus $\sin2x$ means $\sin(2x)$, and $\sin2x\cos2x$ means $(\sin2x)(\cos2x)$. If we really mean $\sin(2x\cos2x)$, ...