(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

\(\int_0^{\pi/5}y\,dx.\)

Find the exact value of

\(\int_0^{\frac{\pi}{2}}\left(\cos3x+4\sin2x+1\right)\,dx.\)

The equation of a curve is

\(y=x\sin x.\)

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

(b) Find the equation of the tangent to the curve at \(x=\frac{\pi}{2}\) in the form \(y=mx+c\).

(c) Use your answer to part (a) to find

\(\int x\cos x\,dx.\)

(d) Evaluate

\(\int_0^{\frac{\pi}{4}}x\cos x\,dx,\)

giving your answer correct to 2 significant figures.

A curve is such that

\(\frac{d^2y}{dx^2}=\sin\left(6x-\frac{\pi}{2}\right).\)

Given that \(\frac{dy}{dx}=\frac12\) at the point \(\left(\frac{\pi}{4},\frac{13\pi}{12}\right)\) on the curve, find the equation of the curve.

(a) Find \(\displaystyle\int \frac{x^2(x^6+1)}{x^6}\,dx\).

(b) (i) Find \(\displaystyle\int \cos(4\theta-5)\,d\theta\).

(ii) Hence evaluate \(\displaystyle\int_{1.25}^{2}\cos(4\theta-5)\,d\theta\).