(a) Integrate the following with respect to \(x\).
(i) \(\mathrm{e}^{5 x-2}\)
(ii) \(\frac{1}{4-3 x}\) where \(x\lt \frac{4}{3}\)
(b) Show that \(\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \sec ^{2}\left(\frac{1}{2} x\right) \mathrm{d} x=2\left(1-\frac{\sqrt{3}}{3}\right)\).

(a) Show that \(\displaystyle \frac{1}{2x-1}+\frac{4}{(2x-1)^2}\) can be written as \(\displaystyle \frac{2x+3}{(2x-1)^2}\).

(b) Find \(\displaystyle \int_2^5 \frac{2x+3}{(2x-1)^2}\,dx\), giving your answer in the form \(a+\ln b\), where \(a\) and \(b\) are constants.

(a) Show that

\(\frac{6}{2+3x}+\frac{4}{(x+1)^2}-\frac{2}{x+1}\)

can be written as

\(\frac{14x+10}{(2+3x)(x+1)^2}.\)

(b) Hence find the exact value of

\(\int_0^2 \frac{14x+10}{(2+3x)(x+1)^2}\,dx.\)

Give your answer in the form \(p+\ln q\), where \(p\) and \(q\) are rational numbers.

(a) Find

\(\frac{d}{dx}\left(x^2e^{3x}\right).\)

(b)

(i) Find

\(\frac{d}{dx}(3x^2+4)^{\frac13}.\)

(ii) Hence find

\(\int_0^2 x(3x^2+4)^{-\frac23}\,dx.\)

(a) Find

\(\int (e^{x+1})^3\,dx.\)

(b)

(i) Differentiate, with respect to \(x\),

\(y=x\sin4x.\)

(ii) Hence show that

\(\int_{\frac{\pi}{4}}^{\frac{\pi}{3}}4x\cos4x\,dx =\frac18-\frac{\pi\sqrt3}{6}.\)

Giving your answer in its simplest form, find the exact value of

(a) \(\displaystyle \int_0^4 \frac{10}{5x+2}\,dx\),

(b) \(\displaystyle \int_0^{\ln2}\left(e^{4x+2}\right)^2\,dx\).

(a)(i) Given that

\(\mathrm{f}(x)=\frac1{\cos x},\)

show that

\(\mathrm{f}'(x)=\tan x\operatorname{sec}x.\)

(ii) Hence find

\(\int\left(3\tan x\operatorname{sec}x-\sqrt[4]{e^{3x}}\right)\,dx.\)

(b) Given that

\(\int_2^5\frac{p}{px+10}\,dx=\ln2,\)

find the value of the positive constant \(p\).

(a) Given that

\(\int_1^a\left(\frac1x-\frac{1}{2x+3}\right)\,dx=\ln3,\)

where \(a\gt 0\), find the exact value of \(a\), giving your answer in simplest surd form.

(b) Find the exact value of

\(\int_0^{\pi/3}\left(\sin\left(2x+\frac{\pi}{3}\right)-1+\cos2x\right)\,dx.\)

The diagram shows the curve \(y=4+2\cos3x\) intersecting the line \(y=5\) at the points \(P\) and \(Q\).

(i) Find, in terms of \(\pi\), the \(x\)-coordinate of \(P\) and of \(Q\).

(ii) Find the exact area of the shaded region.

(a) Given that \(\int_0^a e^{2x}\,dx=50\), find the exact value of \(a\).

(b) A curve is such that \(\frac{dy}{dx}=3-2\cos5x\). The curve passes through \(\left(\frac{\pi}{5},\frac{8\pi}{5}\right)\).

(i) Find the equation of the curve.

(ii) Find \(\int y\,dx\) and hence evaluate \(\int_{\pi/2}^{\pi}y\,dx\).

(a) Find \(\displaystyle\int \sqrt[3]{2x-1}\,dx\).

(b)(i) Find \(\displaystyle\int \sin4x\,dx\).

(b)(ii) Hence evaluate \(\displaystyle\int_{\pi/8}^{\pi/4}\sin4x\,dx\).

(c) Show that \(\displaystyle\int_0^{\ln8} e^{x/3}\,dx=3\).

(i) Find \(\dfrac{d}{dx}(5x^2+4)^{3/2}\).

(ii) Hence find \(\displaystyle\int x(5x^2+4)^{1/2}\,dx\).

Given that \(\displaystyle\int_0^a x(5x^2+4)^{1/2}\,dx=\dfrac{19}{15}\),

(iii) find the value of the positive constant \(a\).

(i) Find \(\dfrac{d}{dx}(5x^2-125)^{2/3}\).

(ii) Using your answer to part (i), find \(\displaystyle\int x(5x^2-125)^{-1/3}\,dx\).

(iii) Hence find \(\displaystyle\int_6^{10}x(5x^2-125)^{-1/3}\,dx\).