(i) Find dy/dx if x = a(θ − sin θ), y = a(1 + sin θ). (1)
(ii) If eʸ(x + 1) = 1, show that d²y/dx² = (dy/dx)². (2)
(iii) Find ∫ (from 0 to π) x/(1 + sin x) dx. (3)

Question

Accepted Answer

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> [!TLDR]
> (i) Differentiate x and y separately with respect to the parameter θ and divide. (ii) Solve for y explicitly from the implicit equation, then differentiate twice. (iii) Use the property ∫₀^a f(x)dx = ∫₀^a f(a−x)dx to convert the integral of x times something into a plain integral.
>
> **(i) $x=a(	heta-\sin	heta)$, $y=a(1+\sin	heta)$.**
> $$\frac{dx}{d	heta} = a(1-\cos	heta), \qquad \frac{dy}{d	heta} = a\cos	heta.$$
> $$\frac{dy}{dx} = \frac{dy/d	heta}{dx/d	heta} = \frac{a\cos	heta}{a(1-\cos	heta)} = \frac{\cos	heta}{1-\cos	heta}.$$
>
> **(ii) $e^y(x+1)=1$. Show $\dfrac{d^2y}{dx^2}=\left(\dfrac{dy}{dx}ight)^2$.**
>
> From the given relation, $e^y = \dfrac{1}{x+1}$, so taking natural log of both sides: $y = -\ln(x+1)$.
>
> Differentiate: $\dfrac{dy}{dx} = -\dfrac{1}{x+1}$.
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> Differentiate again: $\dfrac{d^2y}{dx^2} = \dfrac{1}{(x+1)^2}$ (since $\dfrac{d}{dx}\left[-(x+1)^{-1}ight] = (x+1)^{-2}$).
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> Now compute $\left(\dfrac{dy}{dx}ight)^2 = \left(-\dfrac{1}{x+1}ight)^2 = \dfrac{1}{(x+1)^2}$.
>
> Since both equal $\dfrac{1}{(x+1)^2}$, we get $\dfrac{d^2y}{dx^2}=\left(\dfrac{dy}{dx}ight)^2$, as required. $\blacksquare$
>
> **(iii) $\displaystyle\int_0^\pi \frac{x}{1+\sin x}\,dx$.**
>
> Let $I = \displaystyle\int_0^\pi \frac{x}{1+\sin x}\,dx$. Using the property $\displaystyle\int_0^a f(x)dx = \int_0^a f(a-x)dx$ with $a=\pi$, and noting $\sin(\pi-x)=\sin x$:
> $$I = \int_0^\pi \frac{\pi-x}{1+\sin(\pi-x)}\,dx = \int_0^\pi\frac{\pi-x}{1+\sin x}\,dx.$$
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> Adding this to the original expression for $I$:
> $$2I = \int_0^\pi\frac{x}{1+\sin x}dx+\int_0^\pi\frac{\pi-x}{1+\sin x}dx = \int_0^\pi\frac{\pi}{1+\sin x}dx = \pi\int_0^\pi\frac{dx}{1+\sin x}.$$
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> Compute $J=\displaystyle\int_0^\pi\frac{dx}{1+\sin x}$. Multiply numerator and denominator by $(1-\sin x)$: …

Order	Leibniz	Lagrange	Newton
1st	$\frac{dy}{dx}$	$f'(x)$	$\dot{y}$
2nd	$\frac{d^2y}{dx^2}$	$f''(x)$	$\ddot{y}$
$n$ th	$\frac{d^ny}{dx^n}$	$f^{(n)}(x)$	—

Q.(i) Find dy/dx if x = a(θ − sin θ), y = a(1 + sin θ).

Concept understanding — Successive Differentiation

Successive Differentiation — Repeated Slopes

The definition and notation

Seeing the pattern

Q.(i) Find dy/dx if x = a(θ − sin θ), y = a(1 + sin θ).

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