Further Integration (Cambridge (CIE) A Level Maths: Pure 3): Exam Questions

Integrate

   $\int \frac{1}{2x+3}dx$

Find an expression for $y$given that

   $\frac{dy}{dx}=3e^{3x+1}$

Find the integral

   $\int \sin xdx$

Use calculus to evaluate

      ${\int }_1^4\frac{1}{x}dx$

Find an expression for $y$ given that

   $y=\int 7e^{7x}dx$

Integrate

   $\int \cos 2xdx$

Use calculus to find the value of

   ${\int }_0^2{\left(3x-1\right)}^{3}dx$

Find an expression for $y$ given that

   $\frac{dy}{dx}=e^{5x}$

Use a suitable substitution to find

$\int -15\sin \left(5x-2\right)dx$

Find an expression for $\mathrm{f}\left(x\right)$ given that

   $\mathrm{f}\left(x\right)=\int \sin \left(2x+1\right)dx$

Find an expression for $y$ given that

   $\frac{dy}{dx}=3\cos \left(3x+5\right)$

Use calculus to find the exact value of

${\int }_{\frac{7\mathrm{\pi }}{6}}^{\frac{4\mathrm{\pi }}{3}}se{c}^2\left(x-\mathrm{\pi }\right)dx$

Use calculus and the substitution $u=x+4$ to show that

      ${\int }_1^2\frac{x}{x+4}dx=1+4\mathrm{In}\frac{5}{6}$

Given that

      $\cos 2\theta \equiv 2{\cos }^2\theta -1$

use calculus to find the exact value of

      ${\int }_{\frac{\mathrm{\pi }}{4}}^{\frac{\mathrm{\pi }}{2}}{\cos }^2\theta d\theta$

Find

   $\int \sqrt{1+{\cot }^2}xdx$

i) Given that  $\mathrm{f}\left(x\right)=2x^3+4x,$ find  $\mathrm{f}\text{'}\left(x\right).$

ii) Hence, or otherwise, find

      $\int \frac{3x^2+2}{2x^3+4x}dx$

Use the identity 

      $\sin 2A\equiv 2\sin A\cos A$

to show that

   $4\sin \frac{\theta }{2}\cos \frac{\theta }{2}\equiv 2\sin \theta$

Hence, or otherwise, find the integral

   $\int 4\sin \frac{\theta }{2}\cos \frac{\theta }{2}d\theta$

Show that

   ${\int }_2^{5}4e^{2x-4}dx=2\left(e^6-1\right)$

Use integration by parts to find, in terms of $e$, the exact value of

   ${\int }_0^1(5x-4)e^{3x}dx$

Show that

         $\frac{11}{(2x-3)(x+4)}$

can be written in the form

         $\frac{A}{2x-3}+\frac{B}{x+4}$

Hence find

$\int \frac{11}{(2x-3)(x+4)}dx$

writing your answer as a single logarithm.

Use calculus to find the value of

      ${\int }_2^4\frac{1}{4+9x^2}dx$

giving your answer to three significant figures.

Given that the graph of $y=\mathrm{f}\left(x\right)$  passes through the point (0, 4) and that

      $\mathrm{f}\text{'}\left(x\right)=\frac{12x^2+7}{2\left(4x^3+7x+4\right)}$

Find $\mathrm{f}\left(x\right)$

Explain why $x=-\frac{1}{2}$  must be excluded from the domain of $\mathrm{f}\left(x\right).$

Show that

      $\int \tan xdx=\ln \left|secx\right|+c$

where $c$ is a constant.

Use calculus to find  
        $\int \left(x^2+5-\frac{3}{x^2}\right)dx$

Write down  
        $\int 3e^{3x}dx$

Find the integral

      $\int 5(e^{5x}-e^{-5x})dx$

Find an expression for $y$ given that

      $y=\int \left(\sin x+\cos x\right)dx$

Use calculus to evaluate

         ${\int }_{-8}^{-2}\frac{1}{x}dx$

giving your answer as a single logarithm.

Integrate

      $\int 2\sin x\cos xdx$

Use calculus to find the value of

      ${\int }_1^3{\left(4x+1\right)}^{5}dx$

Use a suitable substitution to find the following

      $\int 8x\sin (3x^2+1)dx$

Use the substitution $u=2+\ln x$ to show that

      $\int \frac{1}{x{\left(2+\ln x\right)}^3}dx=-\frac{1}{2{\left(\ln x+2\right)}^2}+c$

where $c$ is the constant of integration.

Show that, for $\theta \ne k\pi$ (where $k$ is an integer),

         $\frac{2-2{\cos }^2\theta }{\sin 2\theta }=\tan \theta$

Use calculus and your result from part (a) to show that

         ${\int }_{\frac{\mathrm{\pi }}{6}}^{\frac{\mathrm{\pi }}{3}}\frac{2-2{\cos }^2\theta }{\sin 2\theta }d\theta =\frac{1}{2}\ln 3$

Use calculus to find the exact value of

         ${\int }_{\frac{\mathrm{\pi }}{2}}^{\frac{5\mathrm{\pi }}{6}}\frac{2\cos x}{1-\cos 2x}dx.$

Find an expression for $\mathrm{f}\left(x\right)$ given that

         $\mathrm{f}\left(x\right)=\int \frac{5}{2x+3}dx$

Find an expression for $y$ given that

      $\frac{dy}{dx}=2e^{3x-5}-5e^{5x}$

Show that

         ${\sin }^2\left(3x+2\right)\equiv \frac{1}{2}(1-\cos (6x+4\left)\right)$

Hence, or otherwise, find an expression for

      $\int {\sin }^2\left(3x+2\right)dx$

Write  $e^{2x}\left(1+e^{2x}+e^2\right)$  in the form  $e^{\mathrm{f}\left(x\right)}+e^{g\left(x\right)}+e^{h\left(x\right)},$where $\mathrm{f}\left(x\right)$, $\mathrm{g}\left(x\right)$ and $\mathrm{h}\left(x\right)$are all linear functions of $x$.

Hence show that

        ${\int }_0^1{e}^{2x}\left(1+e^{2x}+e^2\right)dx=\frac{3}{4}\left(e^4-1\right)$

Use calculus to show that

        ${\int }_0^1\frac{3x{e}^{-3x^2}}{3-2e^{-3x^2}}dx=\frac{1}{4}\ln (3-2e^{-3}).$

Use integration by parts to find

         $\int \left(2x^2-1\right)e^{x}dx$

Show that

         $\int \ln xdx=x\ln x-x+c$ 

where $c$ is the constant of integration.

Express

         $\frac{x^2-4x+7}{\left(x-1\right){\left(x-3\right)}^2}$

as partial fractions.

Hence, or otherwise, find

         $\int \frac{x^2-4x+7}{\left(x-1\right){\left(x-3\right)}^2}dx$ 

Show that

            ${\int }_0^{\sqrt{\frac{3}{2}}}\frac{4}{9+2x^2}dx=\frac{\mathrm{\pi }\sqrt{2}}{9}$

Given that $\mathrm{f}\text{'}\left(x\right)=2\tan 3x$and that  $\mathrm{f}\left(0\right)=\frac{2}{3}$ show that

            $\mathrm{f}\left(x\right)=\frac{2}{3}(1+\ln \left|sec3x\right|)$

Given

      $\mathrm{f}\left(x\right)=\int \frac{3(8x^3+3x)}{2(4x^4+3x^2+5)}dxx\in \mathrm{ℝ}$

and that the graph of  $y=\mathrm{f}\left(x\right)$  passes through the point (1, ln 144 ) find $\mathrm{f}\left(x\right)$.

Use calculus to find the value of

      ${\int }_2^6\left(\frac{1}{x}+\frac{4}{2x}+\frac{3}{3x-2}\right)dx$

giving your answer in the form  $p\ln p+q\ln q$  ,  where $p$ and $q$ are prime numbers to be found.

Find an expression for $y$given that

      $\frac{dy}{dx}=5{\cos }^{2}4x\sin 4x$

Integrate

      $\int 3x{\left(5x^2+4\right)}^{4}dx$

Find an expression for $y$ given that

         $y=\int 6x^2{e}^{x^3}dx$

Integrate

         $\int \left(16-32x\right)\sin {\left(4x-2\right)}^{2}dx$

Use calculus and the substitution  $x=\cos \theta$   to find the exact value of

      ${\int }_{\frac{1}{\sqrt{2}}}^{\frac{\sqrt{3}}{2}}\frac{1}{\sqrt{1-x^2}}dx$

Find

         $\int {\left(2\tan x+3secx\right)}^{2}dx$

Show that

         $\int \tan kxdx=\frac{1}{k}\ln \left|seckx\right|+c$

where $k$ is a constant, and $c$ is the constant of integration.

Use calculus to find the exact value of

         ${\int }_{\frac{\mathrm{\pi }}{18}}^{\frac{\mathrm{\pi }}{9}}\frac{\cos e{c}^{2}3\theta }{3cot3\theta }d\theta$

writing your answer in the form  $alnb$ , where $a$ and $b$ are rational numbers to be found.

Show that

$\left(\cos \left(\theta +\frac{\mathrm{\pi }}{8}\right)+\sin \left(\theta +\frac{\mathrm{\pi }}{8}\right)\right)\left(\sin \left(\theta +\frac{\mathrm{\pi }}{8}\right)-\cos \left(\theta +\frac{\mathrm{\pi }}{8}\right)\right)\equiv -\cos \left(2\theta +\frac{\mathrm{\pi }}{4}\right)$

Hence, or otherwise, find the exact value of

      ${\int }_0^{\frac{\mathrm{\pi }}{8}}\left({\sin }^2\left(\theta +\frac{\mathrm{\pi }}{8}\right)-{\cos }^2\left(\theta +\frac{\mathrm{\pi }}{8}\right)\right)d\theta$

Show that

   ${\int }_0^1{e}^{ax+b}dx=e^b\left(\frac{e^a-1}{a}\right)$

where $a$ and $b$ are constants, and $a\ne 0.$

Using your working from part (a), or otherwise, evaluate  

      ${\int }_0^c{e}^{ax+b}dx$

giving your answer in terms of $a,b$ and $c$, where $a,b$ and $c$ are constants, and $a\ne 0$.

Show that

      $\left(1+\cot \left(2\theta +\frac{\mathrm{\pi }}{4}\right)\right)\left(1-\cot \left(2\theta +\frac{\mathrm{\pi }}{4}\right)\right)\equiv 2-\cos e{c}^2\left(2\theta +\frac{\mathrm{\pi }}{4}\right)$

Hence, or otherwise, find an expression for $\mathrm{f}\left(\theta \right)$ given that

      $\mathrm{f}\text{'}\left(\theta \right)=\left(2+2\cot \left(2\theta +\frac{\mathrm{\pi }}{4}\right)\right)\left(2-2\cot \left(2\theta +\frac{\mathrm{\pi }}{4}\right)\right)$