5 votes
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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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  • 204k
5 votes

Simpler proof that $y^3[d^2y/dx^2]$ is a constant if $y^2=ax^2+bx+c$?

Differentiating both sides once, we have $$ 2y\frac{\mathrm{d}y}{\mathrm{d}x}=2ax+b, $$ and differentiating twice, we reach $$ 2\left(\frac{\mathrm{d}y}{\mathrm{d}x}\right)^2+2y\frac{\mathrm{d^2}y}{\...
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  • 750
3 votes

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

$$\int x^2(x^3-6)^{34} dx$$ but they wanted us to use integration by parts with u=x^3-6. I certainly agree with amwhy's answer. Integration by parts is an unnecessary complication, since the $u = \...
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3 votes

Simpler proof that $y^3[d^2y/dx^2]$ is a constant if $y^2=ax^2+bx+c$?

Using implicit derivation: $$ (y^2)^\prime = 2yy^\prime = 2ax+b \Rightarrow y^\prime=\frac{2ax+b}{2y} $$ $$ (y^2)^{\prime\prime} = 2(y^\prime)^2+2yy^{\prime\prime} = 2a \Rightarrow y^{\prime\prime}=\...
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2 votes

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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  • 14.1k
2 votes

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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  • 1,919
2 votes
Accepted

Should I be worried about putting brackets around derivatives when I'm using chain rule?

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 $\sin(2x\cos2x)$, ...
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  • 16.4k

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