(a) Given that \(y=x\cos 2x\), find \(\frac{\mathrm{d}y}{\mathrm{d}x}\).

(b) Hence find \(\int x\sin 2x\,\mathrm{d}x\).

(a) Differentiate \(\dfrac{\sin x+\cos x}{\mathrm{e}^{1-3x}}\) with respect to \(x\).

(b) Find \(\displaystyle\int(1+\tan^2 3x)\,\mathrm{d}x\).

It is given that \(y=x\mathrm e^{3x+2}\).

(a) Find \(\frac{\mathrm dy}{\mathrm dx}\).

(b) Hence find \(\int x\mathrm e^{3x+2}\,\mathrm dx\).

(a) Given that \(y=x^{3} \ln x\), find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\). (b) Hence find \(\int_{1}^{2} 3 x^{2} \ln x \mathrm{~d} x\), giving your answer in the form \(\ln a+b\), where \(a\) is an integer and \(b\) is a rational number.

(a) Given that \(y=\tan x-x\), find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\). Write your answer in terms of \(\tan x\). (b) Hence find \(\int_{0}^{\frac{\pi}{4}} \tan ^{2} x \mathrm{~d} x\). Give your answer in exact form.

(a) It is given that \(y=\frac{\ln 3 x}{x^{2}}\) for \(x\gt 0\). Find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\). Give your answer in the form \(\frac{A+B \ln 3 x}{x^{3}}\), where \(A\) and \(B\) are integers. (b) Hence find \(\int \frac{\ln 3 x}{x^{3}} \mathrm{~d} x\).

Given that

\(y=(3x+1)^2\ln(3x+1),\)

(a) find \(\frac{\mathrm dy}{\mathrm dx}\),

(b) hence find \(\displaystyle \int (3x+1)\ln(3x+1)\,\mathrm dx\).

(a) Show that

\(\frac{2}{2x+3}-\frac{1}{x-1}+\frac{1}{(x-1)^2}\)

can be written as

\(\frac{8-3x}{(x-1)^2(2x+3)}.\)

(b) Find

\(\int_2^a \frac{8-3x}{(x-1)^2(2x+3)}\,dx,\)

where \(a\gt2\). Give your answer in the form \(c+\ln d\), where \(c\) and \(d\) are functions of \(a\).

(a) Differentiate \(x\ln x-2x\) with respect to \(x\), simplifying your answer.

(b) A curve is such that

\(\displaystyle \frac{d^2y}{dx^2}=\left(\frac{x+1}{\sqrt{x}}\right)^2\).

At the point \(\left(\mathrm{e},\frac{\mathrm{e}^3}{6}+\mathrm{e}^2\right)\), the gradient of the curve is \(\frac{\mathrm{e}^2}{2}+2\mathrm{e}\). Using your answer to part (a), find the exact equation of the curve.

A curve is such that \(\displaystyle \frac{d^2y}{dx^2}=\left(\frac{\sqrt{x}+1}{\sqrt[4]{x}}\right)^2\). Given that the gradient of the curve is \(\frac43\) at the point \((1,-1)\), find the equation of the curve.

The curve has equation

\(y=kxe^{-2x},\)

where \(k\) is a constant.

(a) Find \(\frac{dy}{dx}\).

(b) The curve \(y=10xe^{-2x}\) has one stationary point. Find the coordinates of this stationary point.

(c) Use your answer to part (a) to find \(\int 4xe^{-2x}\,dx\).

(d) Hence find the exact value of \(\int_0^1 4xe^{-2x}\,dx\).

(a)

(i) Find

\(\int\frac1{(10x-1)^6}\,dx.\)

(ii) Find

\(\int\frac{(2x^3+5)^2}{x}\,dx.\)

(b)

(i) Differentiate \(y=\tan(3x+1)\) with respect to \(x\).

(ii) Hence find

\(\int_{\pi/12}^{\pi/10}\left(\frac{\operatorname{sec}^2(3x+1)}2-\sin x\right)\,dx.\)

(a)

(i) Find

\(\int \sin\left(\frac{\phi+\pi}{3}\right)\,d\phi.\)

(ii) Find

\(\int(5\sin^2\theta+5\cos^2\theta)\,d\theta.\)

(b) Show that

\(\int_1^e\left(\left(1+\frac1x\right)^2-1\right)\,dx=\frac{3e-1}{e}.\)

A curve is such that \(\dfrac{d^2y}{dx^2}=4e^{2x}+3\). When \(x=0\), \(y=-5\) and \(\dfrac{dy}{dx}=10\).

(i) Find the equation of the curve.

(ii) Find the equation of the normal to the curve at the point where \(x=\dfrac14\).

(i) Given that \(\displaystyle y=\frac{\ln x}{x^2}\), find \(\displaystyle \frac{dy}{dx}\).

(ii) Find the coordinates of the stationary point on the curve \(\displaystyle y=\frac{\ln x}{x^2}\).

(iii) Using your answer to part (i), find \(\displaystyle \int \frac{\ln x}{x^3}\,dx\).

(iv) Hence evaluate \(\displaystyle \int_1^2 \frac{\ln x}{x^3}\,dx\).

(i) Differentiate \(x^4\sqrt{\sin x}\) with respect to \(x\).

(ii) Hence find

\(\int\left(x+\frac{x^4\cos x}{\sqrt{\sin x}}+8x^3\sqrt{\sin x}\right)\,dx.\)

It is given that

\(y=(10x+2)\ln(5x+1).\)

(i) Find \(\dfrac{dy}{dx}\).

(ii) Hence show that

\(\int \ln(5x+1)\,dx=\frac{ax+b}{5}\ln(5x+1)-x+c,\)

where \(a\) and \(b\) are integers and \(c\) is a constant of integration.

(iii) Hence find

\(\int_0^{1/5}\ln(5x+1)\,dx,\)

giving your answer in the form \(\dfrac{d+\ln f}{5}\), where \(d\) and \(f\) are integers.

(i) Show that

\(\frac{d}{dx}\left[0.4x^5\left(0.2-\ln(5x)\right)\right]=kx^4\ln(5x),\)

where \(k\) is an integer to be found.

(ii) Express \(\ln(125x^3)\) in terms of \(\ln(5x)\).

(iii) Hence find \(\int x^4\ln(125x^3)\,dx\).

(i) Find \(\dfrac{d}{dx}(x\ln x)\).

(ii) Hence find \(\displaystyle\int\ln x\,dx\).

(iii) Hence, given that \(k\gt 0\), show that \(\displaystyle\int_k^{2k}\ln x\,dx=k(\ln4k-1)\).

(i) Show that

\(\frac{d}{dx}\left(\frac{\ln x}{x^3}\right)=\frac{1-3\ln x}{x^4}.\)

(ii) Find the exact coordinates of the stationary point of the curve \(y=\dfrac{\ln x}{x^3}\).

(iii) Use the result from part (i) to find \(\displaystyle\int \dfrac{\ln x}{x^4}\,dx\).