Given that \(\int_{0}^{2 a+1} \frac{8}{4 x+3} \mathrm{~d} x=\ln 16\), find the exact value of the constant \(a\).

(a) Find \(\int_{2}^{4}(5 x-2)^{-\frac{2}{3}} \mathrm{~d} x\), giving your answer in exact form. (b) Find \(\int_{0}^{\frac{1}{2}}\left(\frac{4}{2 x+1}+\frac{8}{(2 x+1)^{2}}\right) \mathrm{d} x\), giving your answer in the form \(a+\ln b\), where \(a\) and \(b\) are integers.

Find \(\int_{2}^{4}\left(\frac{2}{2 x-3}-\frac{3}{(3 x-5)^{2}}\right) \mathrm{d} x\), giving your answer in exact form.

(a) Evaluate \(\int_{\frac{\pi}{3}}^{\frac{\pi}{2}} \cos \frac{x}{4} \mathrm{~d} x\). You must show all your working. (b) Find \(\int\left(\frac{1}{4 x-3}+\frac{1}{x^{3}}\right) \mathrm{d} x\).

(a) Find

\(\displaystyle \int\left(4x+5-\frac1{2x+3}\right)\,\mathrm dx.\)

(b) Hence find the exact value of

\(\displaystyle \int_1^3\left(4x+5-\frac1{2x+3}\right)\,\mathrm dx,\)

simplifying your answer.

Find

\(\int_0^a\left(\frac{2}{x+1}-\frac{1}{x+2}\right)\,dx,\)

where \(a\) is a positive constant. Give your answer, as a single logarithm, in terms of \(a\).

(a) Show that

\(\frac{1}{2x+1}-\frac{1}{(2x+1)^2}+\frac{4}{4x-1} =\frac{24x^2+14x+4}{(2x+1)^2(4x-1)}.\)

(b) Hence find

\(\int_{\frac12}^{1}\frac{24x^2+14x+4}{(2x+1)^2(4x-1)}\,dx,\)

giving your answer in the form \(\frac12\ln p+q\), where \(p\) and \(q\) are rational numbers.

It is given that

\(\int_1^a\left(\frac{3}{3x+2}-\frac{2}{2x+1}-\frac1x\right)\,dx=\ln\frac15,\)

where \(a\gt 1\). Find the exact value of \(a\).

Given that

\(\int_1^a\left(\frac2{2x+3}+\frac3{3x-1}-\frac1x\right)\,dx=\ln2.4\)

and that \(a\gt 1\), find the value of \(a\).

(a) Show that

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

can be written as

\(\frac{12x}{4x^2-9}.\)

(b) Hence find

\(\int \frac{12x}{4x^2-9}\,dx,\)

giving your answer as a single logarithm and an arbitrary constant.

(c) Given that

\(\int_2^a \frac{12x}{4x^2-9}\,dx=\ln(5\sqrt5),\)

where \(a\gt2\), find the exact value of \(a\).

The gradient of the normal to a curve at the point \((x,y)\) is \(\dfrac{x}{x+1}\). The curve passes through \((1,4)\).

(a) Show that the equation of the curve is

\(y=5-\ln x-x.\)

(b) Find the equation of the tangent to the curve at the point where \(x=3\), giving your answer in terms of \(\ln3\).

Find the exact value of

\(\int_2^4\frac{(x+1)^2}{x^2}\,dx.\)