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International A LevelMaths: Pure 1EdexcelExam Questions5. Integration5.1 Integration

Integration (Edexcel International A Level Maths: Pure 1): Exam Questions

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1
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Integrate

(i)
2 x comma

(ii)
6 x squared comma

(iii)
1 half x to the power of 1 half end exponent. 

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2
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Use calculus to find

integral space open parentheses 3 x to the power of 1 half end exponent space plus space 2 x to the power of negative 1 half end exponent close parentheses space d x.

 

 

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3a
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Show that

fraction numerator 3 x cubed space plus space 4 x to the power of 6 over denominator x squared end fraction

can be written as 3 x to the power of a space plus space 4 x to the power of b comma where a space and space b are constants to be found.

 

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3b
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Hence find

integral space fraction numerator 3 x to the power of 2 space end exponent plus space 4 x to the power of 6 over denominator x squared end fraction space d x.

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4a
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Integrate 5 x to the power of 4 plus 6 x squared plus 2 x plus 3.

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4b
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Given that straight f space open parentheses straight x close parentheses space equals space integral space open parentheses 5 straight x to the power of 4 space end exponent plus space 6 straight x to the power of 2 space end exponent plus space 2 straight x space plus space 3 close parentheses space dx  and that the graph of y space equals space straight f space open parentheses straight x close parentheses passes through the point open parentheses 1 space comma space 10 close parentheses comma find an expression for straight f space open parentheses straight x close parentheses in terms of x only. 

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5
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The curve C, described by the integral

y space equals space integral space open parentheses 2 x to the power of 3 space end exponent minus x close parentheses space d x comma

passes through the point open parentheses 2 space comma space minus 2 close parentheses.

Show that 2 y equals x to the power of 4 minus x squared minus 16.

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This topic pack contains questions on the following:

  • Fundamental Theorem of Calculus

  • Integrating Powers of x

1
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Use calculus to find

 integral left parenthesis 3 x squared plus 5 x plus 3 right parenthesis d x

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2
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Given y equals 2 t squared plus t to the power of negative 1 half end exponent minus 3 t to the power of negative 4 end exponent comma findintegral space y space d t.

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3
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Find the equation of the curve passing through the point (4, 64) and given by

y equals integral left parenthesis 3 minus 2 x plus 6 x squared right parenthesis d x

 

                       

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4a
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Show that 


left parenthesis 3 minus 2 x right parenthesis squared equals 9 minus 12 x plus 4 x squared

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4b
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Hence, or otherwise, work out


integral left parenthesis 3 minus 2 x right parenthesis squared space d x

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5a
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Given fraction numerator d squared space y over denominator d x squared end fraction space equals space 30 x and that when x space equals space 1 comma space fraction numerator d y over denominator d x end fraction space equals space 17 comma  show that

fraction numerator d y over denominator d x end fraction space equals space 15 x squared space plus space 2

                   

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5b
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Find an equation for y in terms of x, given that when x space equals space 2 comma space y space equals space 40.

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1a
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Use calculus to find

integral open parentheses x squared plus 5 minus space 3 over x squared close parentheses space d x

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1b
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Use calculus to find

integral open parentheses 4 over x cubed plus space square root of x close parentheses space d x

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2a
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Use calculus to work out the following integrals:

integral open parentheses 1 third x squared space space plus space 3 x to the power of negative 1 half end exponent close parentheses space d x

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2b
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integral fraction numerator open parentheses x space plus space 3 close parentheses squared over denominator square root of x end fraction d x

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

 y equals integral open parentheses 3 minus 2 x plus space 4 over x squared close parentheses d x

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4a
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Show that

left parenthesis 2 minus x right parenthesis cubed equals 8 minus 12 x plus 6 x squared minus x cubed

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4b
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Hence, or otherwise, work out
integral left parenthesis 2 minus x right parenthesis cubed d x

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5
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A function, straight f space open parentheses straight x close parentheses comma spacehas second derivative given by

                        straight f to the power of apostrophe apostrophe left parenthesis x right parenthesis equals 6 left parenthesis x minus 2 right parenthesis.

Given that f left parenthesis 3 right parenthesis equals 20 comma spaceand straight f to the power of apostrophe open parentheses 2 close parentheses space equals space 8 comma spacefind straight f open parentheses straight x close parentheses.

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1
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Use calculus to find

                           integral space open parentheses 2 square root of x space end root plus space 5 x to the power of 1 third end exponent space close parentheses space d x

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2a
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Use calculus to work out the following integrals:

integral space open parentheses 3 x space minus space fraction numerator square root of 4 x space end root space plus space 5 x to the power of 4 over denominator x end fraction close parentheses space d x

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2b
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integral space open parentheses fraction numerator 3 x over denominator square root of x end fraction space plus space open parentheses 2 x plus 1 close parentheses squared close parentheses space straight d x

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3
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Find the equation of the curve passing through the point (4, -8) and given by

y equals integral space open parentheses fraction numerator 2 over denominator square root of x end fraction space minus space x space minus space 3 close parentheses space d x

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4a
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Show that

             open parentheses 3 minus space 1 half x close parentheses cubed space equals space 27 space minus space 27 over 2 x space plus space 9 over 4 x squared space minus 1 over 8 x cubed

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4b
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Hence, or otherwise, work out


                       integral space open parentheses space 2 open parentheses 3 minus space 1 half x close parentheses space close parentheses cubed space d x

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5
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A function,straight f open parentheses straight x close parentheses comma has second derivative given by

                          straight f apostrophe apostrophe space open parentheses straight x close parentheses space equals space 2 space open parentheses 18 straight x space minus space 5 close parentheses.

Given that open parentheses 2 x space minus space 1 close parentheses and open parentheses 3 x space plus space 2 close parentheses are factors of straight f open parentheses straight x close parentheses comma find straight f space open parentheses straight x close parentheses. 

 

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