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<p>A curve is such that&nbsp;\(\dfrac{\mathrm{d}y}{\mathrm{d}x}=2x+\dfrac{k}{(x+1)^2}\), where&nbsp;\(k\) is a constant. Given that the curve passes through the points \((0,2)\) and \((1,5)\), find the value of \(k\) and the equation of the curve.<span class="inline-marks">[7]</span></p>

A curve is such that \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=2x+\dfrac{k}{(x+1)^2}\), where \(k\) is a constant. Given that the curve passes through the points \((0,2)\) and \((1,5)\), find the value of \(k\) and the equation of the curve.

Problems involving Unknowns

Finding the Equation of the Curve

Reciprocal-square differential equation

Integration

<p>A curve is such that&nbsp;\(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{k}{\sqrt{2x+1}}\), where&nbsp;\(k\) is a constant. Given that the curve passes through the points \((0,4)\) and \((4,10)\), find the value of \(k\) and the equation of the curve.<span class="inline-marks">[7]</span></p>

A curve is such that \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=\dfrac{k}{\sqrt{2x+1}}\), where \(k\) is a constant. Given that the curve passes through the points \((0,4)\) and \((4,10)\), find the value of \(k\) and the equation of the curve.

Radical differential equation through two points

<p>A curve is such that&nbsp;\(\dfrac{\mathrm{d}y}{\mathrm{d}x}=k\mathrm{e}^{2x}\), where&nbsp;\(k\) is a constant. Given that the curve passes through the points \((0,1)\) and \((\ln2,7)\), find the value of \(k\) and the equation of the curve.<span class="inline-marks">[7]</span></p>

A curve is such that \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=k\mathrm{e}^{2x}\), where \(k\) is a constant. Given that the curve passes through the points \((0,1)\) and \((\ln2,7)\), find the value of \(k\) and the equation of the curve.

Exponential differential equation through two points

<p>A curve is such that&nbsp;\(\dfrac{\mathrm{d}y}{\mathrm{d}x}=k\sin2x\), where&nbsp;\(k\) is a constant. Given that the curve passes through the points \((0,1)\) and \(\left(\dfrac{\pi}{4},5\right)\), find the value of \(k\) and the equation of the curve.<span class="inline-marks">[7]</span></p>

A curve is such that \(\dfrac{\mathrm{d}y}{\mathrm{d}x}=k\sin2x\), where \(k\) is a constant. Given that the curve passes through the points \((0,1)\) and \(\left(\dfrac{\pi}{4},5\right)\), find the value of \(k\) and the equation of the curve.

Trigonometric differential equation through two points

<ol class="i-list"><li data-list-item-id="amint-q15-question-1" data-list-item-path="amint-q15-question-1"><div class="list-item-paragraph">Show that&nbsp;\(\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin3x-3x)=-6\sin^2\!\left(\dfrac{3x}{2}\right)\).<span class="inline-marks">[3]</span></div></li><li data-list-item-id="amint-q15-question-2" data-list-item-path="amint-q15-question-2"><div class="list-item-paragraph">Without using a calculator, find the values of the constants \(a\), \(b\) and \(c\) for which \(\displaystyle\int_0^{\pi/9}\left[\sin^2\!\left(\dfrac{3x}{2}\right)-\sin3x\right]\,\mathrm{d}x=a\sqrt3+b\pi+c\).<span class="inline-marks">[5]</span></div></li></ol>
<p>The coefficients in the requested exact form are rational numbers.</p>

Show that \(\dfrac{\mathrm{d}}{\mathrm{d}x}(\sin3x-3x)=-6\sin^2\!\left(\dfrac{3x}{2}\right)\).

Without using a calculator, find the values of the constants \(a\), \(b\) and \(c\) for which \(\displaystyle\int_0^{\pi/9}\left[\sin^2\!\left(\dfrac{3x}{2}\right)-\sin3x\right]\,\mathrm{d}x=a\sqrt3+b\pi+c\).

The coefficients in the requested exact form are rational numbers.

Derivatives of Trigonometric Functions

Definite Integration as the Reverse of Differentiation

Exact constants from a trigonometric integral

Differentiation

<p>If&nbsp;\(\displaystyle\displaystyle\int_{0}^{5} t^{2} + s\,\mathrm{d}t = 30\), find the value of \(s\).</p>

If \(\displaystyle\displaystyle\int_{0}^{5} t^{2} + s\,\mathrm{d}t = 30\), find the value of \(s\).

<p>Given that&nbsp;\(\displaystyle\displaystyle\int_{1}^{2} x + t \, \mathrm{d}x = 10\), find the value of \(t\).</p>

Given that \(\displaystyle\displaystyle\int_{1}^{2} x + t \, \mathrm{d}x = 10\), find the value of \(t\).

<p>Given that&nbsp;\(\displaystyle\displaystyle\int_{0}^{s} 2x - 3 \mathrm{~d}x = -\frac{5}{4}\), find the values of \(s\).</p>

Given that \(\displaystyle\displaystyle\int_{0}^{s} 2x - 3 \mathrm{~d}x = -\frac{5}{4}\), find the values of \(s\).

<p>If&nbsp;\(\displaystyle\displaystyle\int_{0}^{t} x - 6 \ \mathrm{d}x = 16\), find the values of \(t\).</p>

If \(\displaystyle\displaystyle\int_{0}^{t} x - 6 \ \mathrm{d}x = 16\), find the values of \(t\).

<p>A curve for which \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^3+a}{x^2}\), where&nbsp;\(a\) is a constant, has a gradient value of \(4\) at the point \((-1, 3)\).</p>
<ol class="i-list"><li><div class="list-item-content"><span class="list-item-text">Find the value of \(a\).</span></div></li><li><div class="list-item-content"><span class="list-item-text">Find the equation of the curve.</span></div></li></ol>

A curve for which \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{x^3+a}{x^2}\), where \(a\) is a constant, has a gradient value of \(4\) at the point \((-1, 3)\).

<p>The curve for which \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{a}{(2x+3)^3} - 1\) where \(a\) is a constant, is such that the tangent to the curve at \((-1, 0)\) is perpendicular to the line \(5y = x + 1\). Find the value of \(a\) and the equation of the curve.</p>

The curve for which \(\frac{\mathrm{d}y}{\mathrm{d}x} = \frac{a}{(2x+3)^3} - 1\) where \(a\) is a constant, is such that the tangent to the curve at \((-1, 0)\) is perpendicular to the line \(5y = x + 1\). Find the value of \(a\) and the equation of the curve.

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