Find the equation of the curve which passes through the point \((2,17)\) and for which
\(\frac{dy}{dx}=4x^3+1.\)
A curve is such that \(\dfrac{dy}{dx}=10e^{5x}+3\). It is given that the curve passes through the point \((0,9)\). Find the equation of the curve.
Find \(y\) in terms of \(x\), given that \(\dfrac{d^2y}{dx^2}=6x+\dfrac2{x^3}\) and that when \(x=1\), \(y=3\) and \(\dfrac{dy}{dx}=1\).
The gradient of the normal to a curve at the point with coordinates \((x,y)\) is given by \(\dfrac{\sqrt x}{1-3x}\).
(i) Find the equation of the curve, given that the curve passes through the point \((1,-10)\).
(ii) Find, in the form \(y=mx+c\), the equation of the tangent to the curve at the point where \(x=4\).
(a) Given that \(y=x^{2} \ln x\), find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\). (b) Hence find \(\int x \ln x \mathrm{~d} x\).
Given that \(\mathrm{f}^{\prime \prime}(x)=(3 x+5)^{-\frac{2}{3}}, \mathrm{f}^{\prime}(1)=6\), and \(\mathrm{f}(1)=20\), find an expression for \(\mathrm{f}(x)\).
The second derivative of a curve is given by
\(\frac{d^2y}{dx^2}=(3x+2)^{-\frac13}.\)
The gradient of the curve is \(4\) at the point \((2,6.2)\).
Find the equation of the curve.
A curve is such that \(\frac{d^2y}{dx^2}=(2x+3)^{-1/2}\). The curve has a gradient of \(5\) at the point where \(x=3\), and passes through the point \(\left(\frac12,-\frac13\right)\).
(i) Find the equation of the curve.
(ii) Find the equation of the normal to the curve at the point where \(x=3\), giving your answer in the form \(ax+by+c=0\), where \(a\), \(b\) and \(c\) are integers.
(i) Find \(\displaystyle\int(7x-10)^{-\frac35}\,dx\).
(ii) Given that \(\displaystyle\int_6^a(7x-10)^{-\frac35}\,dx=\dfrac{25}{14}\), find the exact value of \(a\).
(i) Find \(\dfrac{d}{dx}\left(\dfrac5{3x+2}\right)\).
(ii) Use your answer to part (i) to find \(\displaystyle\int\dfrac{30}{(3x+2)^2}\,dx\).
(iii) Hence evaluate \(\displaystyle\int_1^2\dfrac{30}{(3x+2)^2}\,dx\).
(a) Find \(\int \frac{1}{\sqrt{3 x+2}} \mathrm{~d} x\). (b) Find, in terms of a, \(\int_{0.5}^{a} \mathrm{e}^{(1-2 x)} \mathrm{d} x\).
Find the exact area of the region enclosed by the curve \(y=\mathrm{e}^{2-4 x}\), the \(x\)-axis, the line \(x=-0.25\) and the line \(x=0.5\).
(a) Differentiate \(y=2xe^{4x}\) with respect to \(x\).
(b) Hence find \(\int xe^{4x}\,dx\).
A curve with equation \(y=\mathrm f(x)\) is such that
\(\displaystyle \frac{\mathrm d^2y}{\mathrm dx^2}=6e^{3x}+4x.\)
The curve has a gradient of \(5\) at the point \(\left(0,\frac53\right)\). Find \(\mathrm f(x)\).
The equation of a curve is given by
\(y=xe^{-2x}.\)
(i) Find \(\frac{dy}{dx}\).
(ii) Find the exact coordinates of the stationary point on the curve \(y=xe^{-2x}\).
(iii) Find, in terms of \(e\), the equation of the tangent to the curve \(y=xe^{-2x}\) at the point \(\left(1,\frac1{e^2}\right)\).
(iv) Using your answer to part (i), find \(\int xe^{-2x}\,dx\).
The curve \(y=f(x)\) passes through the point \(\left(\dfrac12,\dfrac72\right)\) and is such that
\(f'(x)=e^{2x-1}.\)
(i) Find the equation of the curve.
(ii) Find the value of \(x\) for which \(f''(x)=4\), giving your answer in the form \(a+b\ln\sqrt2\), where \(a\) and \(b\) are constants.
It is given that
\(\int_{-k}^{k}\left(15e^{5x}-5e^{-5x}\right)\,dx=6.\)
(i) Show that
\(e^{5k}-e^{-5k}=3.\)
(ii) Hence, using the substitution \(y=e^{5k}\), or otherwise, find the value of \(k\).
(a) Show that \(\frac{1+\cot ^{2} \theta}{\cot ^{2} \theta}=\sec ^{2} \theta\). (b) Write down the derivative of \(\tan \theta\) with respect to \(\theta\). (c) Using part (a) and part (b), find the exact value of \(\int_{0}^{\frac{\pi}{3}}\left(\frac{1+\cot ^{2} \theta}{\cot ^{2} \theta}-\sin \theta\right) \mathrm{d} \theta\).
A curve is such that \(\frac{\mathrm{d}^{2} y}{\mathrm{~d} x^{2}}=\cos \left(4 x-\frac{\pi}{4}\right)\). Given that \(\frac{\mathrm{d} y}{\mathrm{~d} x}=\frac{3}{4}\) at the point \(\left(\frac{3 \pi}{16}, \frac{\pi}{4}\right)\) on the curve, find the equation of the curve.
Given that
\(y=\frac{\operatorname{sec}^2 5x-\tan^2 5x}{\operatorname{cosec}5x},\)
show that \(y=a\sin bx\), where \(a\) and \(b\) are integers to be found.
Hence find
\(\int_0^{\pi/5}y\,dx.\)