Using the substitution \(u = \sqrt{x}\), find the exact value of \(\int_{3}^{\infty} \frac{1}{(x+1)\sqrt{x}} \, dx\).
Let \(I = \int_{\frac{1}{4}}^{\frac{3}{4}} \sqrt{\left( \frac{x}{1-x} \right)} \, dx\).
(i) Using the substitution \(x = \cos^2 \theta\), show that \(I = \int_{\frac{1}{6}\pi}^{\frac{1}{3}\pi} 2 \cos^2 \theta \, d\theta\).
(ii) Hence find the exact value of \(I\).
Let \(I = \int_{1}^{4} \frac{(\sqrt{x}) - 1}{2(x + \sqrt{x})} \, dx\).
Using the substitution \(u = \sqrt{x}\), show that \(I = \int_{1}^{2} \frac{u - 1}{u + 1} \, du\).
Let \(I = \int_0^1 \frac{x^5}{(1+x^2)^3} \, dx\).
(i) Using the substitution \(u = 1 + x^2\), show that \(I = \int_1^2 \frac{(u-1)^2}{2u^3} \, du\).
(ii) Hence find the exact value of \(I\).
Let \(I = \int_0^1 \frac{9}{(3 + x^2)^2} \, dx\).
(i) Using the substitution \(x = (\sqrt{3}) \tan \theta\), show that \(I = \sqrt{3} \int_0^{\frac{\pi}{6}} \cos^2 \theta \, d\theta\).
(ii) Hence find the exact value of \(I\).
Use the substitution \(u = 4 - 3 \cos x\) to find the exact value of \(\int_{0}^{\frac{1}{2}\pi} \frac{9 \sin 2x}{\sqrt{(4 - 3 \cos x)}} \, dx.\)
Let \(I = \int_0^1 \frac{\sqrt{x}}{2 - \sqrt{x}} \, dx\).
(i) Using the substitution \(u = 2 - \sqrt{x}\), show that \(I = \int_1^2 \frac{2(2-u)^2}{u} \, du\).
(ii) Hence show that \(I = 8 \ln 2 - 5\).
Find the exact value of \(\int_{0}^{\frac{1}{4}\pi} x \sec^2 x \, dx\).
Show that \(\int_0^{\frac{1}{4}\pi} x^2 \cos 2x \, dx = \frac{1}{32}(\pi^2 - 8)\).
Show that \(\int_{1}^{4} x^{-\frac{3}{2}} \ln x \, dx = 2 - \ln 4\).
(i) Find \(\int \frac{\ln x}{x^3} \, dx\).
(ii) Hence show that \(\int_1^2 \frac{\ln x}{x^3} \, dx = \frac{1}{16}(3 - \ln 4)\).
Showing all necessary working, find the value of \(\int_{0}^{\frac{1}{6}\pi} x \cos 3x \, dx\), giving your answer in terms of \(\pi\).
Find the exact value of \(\int_{0}^{\frac{1}{2}\pi} \theta \sin \frac{1}{2} \theta \, d\theta\).
Find the exact value of \(\int_{0}^{\frac{1}{2}\pi} x^2 \sin 2x \, dx\).
Find the exact value of \(\int_{0}^{\frac{1}{2}} xe^{-2x} \, dx\).
Find the exact value of \(\int_{1}^{4} \frac{\ln x}{\sqrt{x}} \, dx\).
Show that \(\int_{2}^{4} 4x \ln x \, dx = 56 \ln 2 - 12\).
The expression \(f(x)\) is defined by \(f(x) = 3x e^{-2x}\).
(i) Find the exact value of \(f'\left(-\frac{1}{2}\right)\).
(ii) Find the exact value of \(\int_{-\frac{1}{2}}^{0} f(x) \, dx\).
Find the exact value of \(\int_{\frac{1}{3}\pi}^{\pi} x \sin \frac{1}{2}x \, dx\).
Show that \(\int_{0}^{1} (1-x)e^{-\frac{1}{2}x} \, dx = 4e^{-\frac{1}{2}} - 2\).