Integration (Cambridge (CIE) AS Maths: Pure 1): Exam Questions

Use calculus to find

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

Evaluate

      ${\int }_1^5(4x+6x^2)\mathrm{d}x$

Find the equation of the curve passing through the point (4, 64) and given by

        $y=\int \left(3-2x+6x^2\right)\mathrm{d}x$

Show that

${\left(3-2x\right)}^2=9-12x+4x^2$

Hence, or otherwise, work out

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

Given

      ${\int }_k^5\left(2x-1\right)dx=20$

find the value of the positive constant $k.$

Given  $\frac{d^{2}y}{d{x}^2}=30x$ and that when $x=1$ ,$\frac{dy}{dx}=17$,  show that

      $\frac{dy}{dx}=15x^2+2$

Find an equation for y in terms of x, given that when $x=2$, $y=40$.

Use calculus to show that

      ${\int }_1^a\frac{2}{x^3}dx=1-\frac{1}{a^2}$

where $a>1$ is a constant.

By first considering the limit of $\frac{1}{a^2}$ as $a\to \infty$,  determine the value of

      ${\int }_1^{\infty }\frac{2}{x^3}dx$

Use calculus to find the value of

${\int }_4^9\frac{x^2+1}{\sqrt{x}}\mathrm{d}x$

Find the equation of the curve passing through the point (-2, 3) and given by

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

Using the binomial expansion, or otherwise, show that

      ${\left(2-x\right)}^3=8-12x+6x^2-x^3$

Hence, or otherwise, work out

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

Given

${\int }_1^p(1+\frac{1}{x^2})\mathrm{d}x=\frac{15}{4}$

find the value of the constant p, where $p>0$.

A function,$\mathrm{f}\left(x\right)$, has second derivative given by

      $\mathrm{f}\text{'}\text{'}\left(x\right)=6(x-2).$

Given that $\mathrm{f}\left(3\right)=20$, and $\mathrm{f}\text{'}\left(2\right)=8,$ find $\mathrm{f}\left(x\right).$

Use calculus to find the exact value of each of the following improper integrals:

      ${\int }_7^{\infty }\frac{5}{x^2}dx$

      ${\int }_0^{18}\frac{2}{\sqrt{x}}dx$

Use calculus to find

      $\int \left(2\sqrt{x}+5x^{\frac{1}{3}}\right)\mathrm{d}x$

Use calculus to find the value of

${\int }_2^4\frac{x^3+\sqrt[3]{x}}{2\sqrt{x}}dx$

giving your answer correct to 3 significant figures.

Find the equation of the curve passing through the point (4, -8) and given by

      $y=\int \left(\frac{2}{\sqrt{x}}-x-3\right)\mathrm{d}x$

Show that

      ${\left(3-\frac{1}{2}x\right)}^3=27-\frac{27}{2}x+\frac{9}{4}{x}^2-\frac{1}{8}{x}^3$

Hence, or otherwise, work out

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

Given

      ${\int }_q^{4q}5x\sqrt{x}\mathrm{d}x=15066$

find the value of the constant q.

A function, $\mathrm{f}\left(x\right)$, has second derivative given by

       $\mathrm{f}\text{'}\text{'}\left(x\right)=2\left(18x-5\right).$

Given that $(2x-1)$ and $(3x+2)$ are factors of $\mathrm{f}\left(x\right)$, find $\mathrm{f}\left(x\right)$.

Given that

      ${\int }_p^{\infty }\frac{3}{x\sqrt{x}}dx=\sqrt{3}$ 

where $p$ is a real constant, find the value of $p$.

Given that

      ${\int }_0^{50}\frac{q+3x}{\sqrt{x}}dx=750\sqrt{2}$

where $q$ is a real constant, find the value of $q$.