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 \(\displaystyle\int_{1.25}^{2}\cos(4\theta-5)\,d\theta\).
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
\(\int_2^4\frac{(x+1)^2}{x^2}\,dx.\)
(a) Given that \(y=x\cos 2x\), find \(\frac{\mathrm{d}y}{\mathrm{d}x}\).
(b) Hence find \(\int x\sin 2x\,\mathrm{d}x\).
(a) Differentiate \(\dfrac{\sin x+\cos x}{\mathrm{e}^{1-3x}}\) with respect to \(x\).
(b) Find \(\displaystyle\int(1+\tan^2 3x)\,\mathrm{d}x\).
It is given that \(y=x\mathrm e^{3x+2}\).
(a) Find \(\frac{\mathrm dy}{\mathrm dx}\).
(b) Hence find \(\int x\mathrm e^{3x+2}\,\mathrm dx\).
(a) Given that \(y=x^{3} \ln x\), find \(\frac{\mathrm{d} y}{\mathrm{~d} x}\). (b) Hence find \(\int_{1}^{2} 3 x^{2} \ln x \mathrm{~d} x\), giving your answer in the form \(\ln a+b\), where \(a\) is an integer and \(b\) is a rational number.