(i) Prove the identity \(\sin^2 \theta \cos^2 \theta \equiv \frac{1}{8}(1 - \cos 4\theta)\).
(ii) Hence find the exact value of \(\int_{0}^{\frac{1}{3}\pi} \sin^2 \theta \cos^2 \theta \, d\theta\).
(i) Prove the identity \(\cot x - \cot 2x \equiv \csc 2x\).
(ii) Show that \(\int_{\frac{1}{6}\pi}^{\frac{1}{4}\pi} \cot x \, dx = \frac{1}{2} \ln 2\).
(iii) Find the exact value of \(\int_{\frac{1}{6}\pi}^{\frac{1}{4}\pi} \csc 2x \, dx\), giving your answer in the form \(a \ln b\).
(a) Prove that \(\csc 2\theta - \cot 2\theta \equiv \tan \theta\).
(b) Hence show that \(\int_{\frac{1}{4}\pi}^{\frac{3}{4}\pi} (\csc 2\theta - \cot 2\theta) \, d\theta = \frac{1}{2} \ln 2\).
(a) Prove that \(\frac{1 - \cos 2\theta}{1 + \cos 2\theta} \equiv \tan^2 \theta\).
(b) Hence find the exact value of \(\int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi} \frac{1 - \cos 2\theta}{1 + \cos 2\theta} \, d\theta\).
Let \(f(x) = \frac{\cos x}{1 + \sin x}\).
(a) Show that \(f'(x) < 0\) for all \(x\) in the interval \(-\frac{1}{2}\pi < x < \frac{3}{2}\pi\).
(b) Find \(\int_{\frac{1}{6}\pi}^{\frac{1}{2}\pi} f(x) \, dx\). Give your answer in a simplified exact form.
Let \(f(\theta) = \frac{1 - \cos 2\theta + \sin 2\theta}{1 + \cos 2\theta + \sin 2\theta}\).
(i) Show that \(f(\theta) = \tan \theta\).
(ii) Hence show that \(\int_{\frac{\pi}{6}}^{\frac{\pi}{4}} f(\theta) \, d\theta = \frac{1}{2} \ln \frac{3}{2}\).
(i) By first expanding \(\sin(2x + x)\), show that \(\sin 3x \equiv 3 \sin x - 4 \sin^3 x\).
(ii) Hence, showing all necessary working, find the exact value of \(\int_0^{\frac{1}{3}\pi} \sin^3 x \, dx\).
A curve has equation \(y = \frac{3 \cos x}{2 + \sin x}\), for \(-\frac{1}{2}\pi \leq x \leq \frac{1}{2}\pi\).
(i) Find the exact coordinates of the stationary point of the curve.
(ii) The constant \(a\) is such that \(\int_0^a \frac{3 \cos x}{2 + \sin x} \, dx = 1\). Find the value of \(a\), giving your answer correct to 3 significant figures.
(i) Show that \(\frac{2 \sin x - \sin 2x}{1 - \cos 2x} \equiv \frac{\sin x}{1 + \cos x}\).
(ii) Hence, showing all necessary working, find \(\int_{\frac{1}{3}\pi}^{\frac{1}{2}\pi} \frac{2 \sin x - \sin 2x}{1 - \cos 2x} \, dx\), giving your answer in the form \(\ln k\).
Let \(I = \int_0^3 \frac{27}{(9 + x^2)^2} \, dx\).
(a) Using the substitution \(x = 3 \tan \theta\), show that \(I = \int_0^{\frac{\pi}{4}} \cos^2 \theta \, d\theta\).
(b) Hence find the exact value of \(I\).
Use the substitution \(u = 1 + 3 \tan x\) to find the exact value of
\(\int_{0}^{\frac{\pi}{4}} \frac{\sqrt{1 + 3 \tan x}}{\cos^2 x} \, dx.\)
Use the substitution \(u = 3x + 1\) to find \(\int \frac{3x}{3x+1} \, dx\).
Using the substitution \(x = (\sqrt{3}) \tan \theta\), find the exact value of
\(\int_{1}^{3} \frac{1}{\sqrt{3 + x^2}} \, dx,\)
expressing your answer as a single logarithm.
Use the substitution \(u = \\sin 4x\) to find the exact value of \(\int_{0}^{\frac{1}{24}\pi} \cos^3 4x \, dx\).
Let \(I = \int_0^1 \frac{x^2}{\sqrt{(4-x^2)}} \, dx\).
(i) Using the substitution \(x = 2 \sin \theta\), show that \(I = \int_0^{\frac{\pi}{6}} 4 \sin^2 \theta \, d\theta\).
(ii) Hence find the exact value of \(I\).
(i) Use the substitution \(x = 2 \tan \theta\) to show that
\(\int_0^2 \frac{8}{(4+x^2)^2} \, dx = \int_0^{\frac{\pi}{4}} \cos^2 \theta \, d\theta.\)
(ii) Hence find the exact value of
\(\int_0^2 \frac{8}{(4+x^2)^2} \, dx.\)
Let \(I = \int_1^4 \frac{1}{x(4 - \sqrt{x})} \, dx\).
Use the substitution \(u = \sqrt{x}\) to show that \(I = \int_1^2 \frac{2}{u(4-u)} \, du\).
(i) Use the substitution \(x = \sin^2 \theta\) to show that \(\int \sqrt{\left( \frac{x}{1-x} \right)} \, dx = \int 2 \sin^2 \theta \, d\theta\).
(ii) Hence find the exact value of \(\int_0^{\frac{1}{4}} \sqrt{\left( \frac{x}{1-x} \right)} \, dx\).
(i) Use the substitution \(x = \tan \theta\) to show that
\(\int \frac{1-x^2}{(1+x^2)^2} \, dx = \int \cos 2\theta \, d\theta.\)
(ii) Hence find the value of
\(\int_0^1 \frac{1-x^2}{(1+x^2)^2} \, dx.\)
Let \(f(x) = \frac{1}{(9-x)\sqrt{x}}\).
(a) Find the \(x\)-coordinate of the stationary point of the curve with equation \(y = f(x)\).
(b) Using the substitution \(u = \sqrt{x}\), show that \(\int_0^4 f(x) \, dx = \frac{1}{3} \ln 5\).