Further Integration (Edexcel International A Level (IAL) Maths: Pure 3): Exam Questions

Find the integral

         $\int \sin x\mathrm{d}x$

Use calculus to evaluate

            ${\int }_1^4\frac{1}{x}\mathrm{d}x$

Find an expression for $y$ given that

            $y=\int 7e^{7x}\mathrm{d}x$

Integrate

      $\int \cos 2x\mathrm{d}x$

Use calculus to find the value of

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

Find an expression for $y$ given that

               $\frac{\mathrm{d}y}{\mathrm{d}x}=e^{5x}$

Given that

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

use calculus to find the exact value of

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

Find

      $\int \sqrt{1+{\cot }^{2}x}\mathrm{d}x$

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

(ii) Hence, or otherwise, find

            $\int \frac{3x^2+2}{2x^3+4x}\mathrm{d}x$

Integrate

      $\int \frac{1}{2x+3}\mathrm{d}x$

Find an expression for given that

               $\frac{\mathrm{d}y}{\mathrm{d}x}=3e^{3x+1}$

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

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

Find an expression for given that

            $\frac{\mathrm{d}y}{\mathrm{d}x}=3\cos \left(3x+5\right)$

Use calculus to find the exact value of

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

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

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

Show that

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

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

      $\int \tan x\mathrm{d}x=\ln \left|secx\right|+c$

where $c$ is a constant.

Find the integral

         $\int 5\left(e^{5x}-e^{-5x}\right)\mathrm{d}x$

Find an expression for $y$ given that

            $y=\int (\sin x+\cos x)\mathrm{d}x$

Use calculus to evaluate

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

giving your answer as a single logarithm.

Integrate

        $\int 2\sin x\cos x\mathrm{d}x$

Use calculus to find the value of

            ${\int }_1^3(4x+1{)}^5\mathrm{d}x$

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

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

Use calculus to find the exact value of

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

Use calculus to show that

            ${\int }_0^1\frac{3x{e}^{-3x^2}}{3-2e^{-3x^2}}\mathrm{d}x=\frac{1}{4}\mathrm{In}\left(3-2e^{-3}\right)$.

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

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

Find an expression for $y$ given that

               $\frac{\mathrm{d}y}{\mathrm{d}x}=2e^{3x-5}-5e^{5x}$

Show that

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

Hence, or otherwise, find an expression for

               $\int {\sin }^2(3x+2)\mathrm{d}x$

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

Hence show that

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

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

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

Given

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

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

Find an expression for $y$ given that

               $\frac{\mathrm{d}y}{\mathrm{d}x}=5{\cos }^{2}4x\sin 4x$

Integrate

      $\int 3x(5x^2+4{)}^4\mathrm{d}x$

Find

   $\int (2\tan x+3\sec x{)}^2\mathrm{d}x$

Show that

      $\int \tan kx\mathrm{d}x=\frac{1}{k}\ln \left|\sec kx\right|+c$

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

Use calculus to find the exact value of

         ${\int }_{\frac{\mathrm{\pi }}{18}}^{\frac{\mathrm{\pi }}{9}}\frac{{\mathrm{cosec}}^{2}3\theta }{3\cot 3\theta }\mathrm{d}\theta$

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

Use calculus to find the value of

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

giving your answer in the form   $p\ln p+q\ln q$,  where $p$ and $q$ are prime 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

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

Show that

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

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

Using your working from part [a], or otherwise, evaluate 

            ${\int }_0^c{e}^{ax+b}\mathrm{d}x$

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

Show that

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

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

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

Show that there are no positive values of $a$ and $b$ that satisfy the equation

               ${\int }_0^1\frac{2a}{ax+b}\mathrm{d}x={\int }_b^{2a}\frac{1}{x}\mathrm{d}x$

Given that

      $\mathrm{f}\text{'}\left(\mathrm{\theta }\right)=\frac{9}{3-3{\sin }^2\left(3\theta -{\displaystyle \frac{\pi }{6}}\right)}$

and also that

         ${\left[\mathrm{f}\left(\frac{\mathrm{\pi }}{6}\right)\right]}^2-{\left[\mathrm{f}\left(\frac{\mathrm{\pi }}{9}\right)\right]}^2=\frac{8}{3}\left(1-\sqrt{3}\right)$

find $\mathrm{f}\left(\theta \right)$.

Given that  $\mathrm{f}\text{'}\left(x\right)=\frac{1}{4}\tan \left(\frac{x}{2}\right)\mathrm{and}\mathrm{f}\left(\frac{2\mathrm{\pi }}{3}\right)=\frac{1}{2}\mathrm{In}6$, show that 

               $\mathrm{f}\left(x\right)=\mathrm{In}\sqrt{\left|3\sec \left(\frac{x}{2}\right)\right|}$