Integration (Edexcel IAL Maths): Exam Questions

Integrate

(i) $2x,$

(ii) $6x^{2},$

(iii) $\frac{1}{2}x^{\frac{1}{2}}.$

Use calculus to find

$\int ( 3 x^{1 \frac{1}{2}} + 2 x^{- 1 \frac{1}{2}} ) d x.$

Show that

$\frac{3x^{3}+4x^{6}}{x^{2}}$

can be written as $3x^{a}+4x^{b},$ where $a\mathrm{and}b$ are constants to be found.

Hence find

$\int \frac{3 x^{2 } + 4 x^{6} }{x^{2}} d x.$

Integrate $5x^{4}+6x^{2}+2x+3.$

Given that $\mathrm{f} ( \mathrm{x} ) = \int ( 5 \mathrm{x}^{4 } + 6 \mathrm{x}^{2 } + 2 \mathrm{x} + 3 ) \text{dx}$ and that the graph of $y = \mathrm{f} ( \mathrm{x} )$ passes through the point $( 1 , 10 ) ,$ find an expression for $\mathrm{f} ( \mathrm{x} )$ in terms of $x$ only.

The curve $C$, described by the integral

$y = \int ( 2 x^{3 } - x ) d x ,$

passes through the point $( 2 , - 2 ).$

Show that $2y=x^{4}-x^{2}-16.$

Use calculus to find

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

Given $y=2t^{2}+t^{-\frac{1}{2}}-3t^{-4},$ find$\int y d t.$

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

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

Show that

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

Hence, or otherwise, work out

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

Given $\frac{d^{2}y}{dx^{2}}=30x$ 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 find

$\int ( x^{2} + 5 - \frac{3}{x^{2}} ) d x$

Use calculus to find

$\int ( \frac{4}{x^{3}} + \sqrt{x} ) d x$

Use calculus to work out the following integrals:

$\int ( 1 \frac{1}{3} x^{2} + 3 x^{- 1 \frac{1}{2}} ) d x$

$\int \frac{( x + 3 )^{2}}{ \sqrt{x} } d x$

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

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

Show that $(2-x)^{3}=8-12x+6x^{2}-x^{3}$

Hence, or otherwise, work out $\int ( 2 - x )^{3} d x$

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

f^{'}'(x)=6(x-2).

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

Use calculus to find

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

Use calculus to work out the following integrals:

$\int ( 3 x - \frac{ \sqrt{4 x } + 5 x^{4} }{x} ) d x$

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

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

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

Show that

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

Hence, or otherwise, work out

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

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

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

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