// Numbas version: exam_results_page_options {"name": "Review Week 2023", "metadata": {"description": "

Exam for Maths 4, Review week 2023

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Integration by parts.

Integrate the following expression:

$f_{(x)}=\\var{a}x+x+\\var{b}x^\\var{c}+sin(\\var{d}x)$

", "advice": "

\\[ f_{(x)}=\\var{a}x+x+\\var{b}x^\\var{c}+sin(\\var{d}x) \\]

Each part of the expression can be integrated separately.

\\[\\int{\\var{a}x}.dx=\\var{a}\\int{x}.dx=\\var{a}\\frac{x^2}{2}\\]

\\[ \\int{x.dx}=\\frac{x^2}{2}\\]

\\[\\int{\\var{b}x^{\\var{c}}}.dx=\\var{b}\\int{x^{\\var{c}}}.dx=\\var{b}\\frac{x^\\var{c+1}}{\\var{c+1}}\\]

For $\\int{sin(\\var{d}x)}.dx$

$let u = \\var{d}x$

$\\frac{du}{dx}=\\var{d} \\therefore dx = \\frac{du}{\\var{d}}$

Sub back in

\\[\\int{sin(u)}.\\frac{du}{\\var{d}}=\\frac{1}{\\var{d}}\\int{sin(u).du}=\\frac{-cos(u)}{\\var{d}}=\\frac{-cos(\\var{d}x)}{\\var{d}} \\]

Recombine and add the constant of integration

\\[ \\var{a}\\frac{x^2}{2}+ \\frac{x^2}{2} + \\var{b}\\frac{x^\\var{c+1}}{\\var{c+1}} + \\frac{-cos(\\var{d}x)}{\\var{d}} +C\\]

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Use $C$ as the constant of integration. Ensure variables and functions are in the numerator of any fractions.

", "answer": "({a}*x^2)/2+x^2/2+({b}x^{c+1})/{c+1}-cos({d}x)/{d}+C", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "caseSensitive": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "x", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "Part Integral 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "Brendan Kennedy", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/20026/"}], "tags": [], "metadata": {"description": "

Integration by parts.

Integrate the following expression:

$f_{(x)} = \\var{a}x^{\\var{c}}+e^{\\var{b}x}+cos(\\var{d}x)$

", "advice": "

\\[ f_{(x)} = \\var{a}x^{\\var{c}}+e^{\\var{b}x}+cos(\\var{d}x)$ \\]

Each part of the expression can be integrated separately.

\\[\\int{\\var{a}x^{\\var{c}}}.dx=\\var{a}\\int{x^{\\var{c}}}.dx=\\var{a}\\frac{x^\\var{c+1}}{\\var{c+1}}\\]

\\[\\int{e^{\\var{b}x}}.dx = \\frac{e^{\\var{b}x}}{\\var{b}} \\]

For $\\int{cos(\\var{d}x)}.dx$

$let u = \\var{d}x$

$\\frac{du}{dx}=\\var{d} \\therefore dx = \\frac{du}{\\var{d}}$

Sub back in

\\[\\int{cos(u)}.\\frac{du}{\\var{d}}=\\frac{1}{\\var{d}}\\int{cos(u).du}=\\frac{sin(u)}{\\var{d}}=\\frac{sin(\\var{d}x)}{\\var{d}} \\]

Recombine and add the constant of integration

\\[ \\var{a}\\frac{x^\\var{c+1}}{\\var{c+1}}+ \\frac{e^{\\var{b}x}}{\\var{b}} + \\frac{sin(\\var{d}x)}{\\var{d}} +C\\]

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Use C as the constant on integration. Ensure variables and functions are in the numerator of any fractions.

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Integration by parts.

Integrate the following expression:

$f(x)=\\var{a}x^{\\var{b}}+\\frac{\\var{c}}{x}+e^{\\var{d}x}$

", "advice": "

\\[ f_{(x)}=\\var{a}x^{\\var{b}}+\\frac{\\var{c}}{x}+e^{\\var{d}x} \\]

Each part of the expression can be integrated separately.

\\[\\int{\\var{a}x^{\\var{b}}}.dx=\\var{a}\\int{x^{\\var{b}}}.dx=\\var{a}\\frac{x^\\var{b+1}}{\\var{b+1}}\\]

\\[ \\int{\\frac{\\var{c}}{x}}.dx=\\var{c}\\int{\\frac{1}{x}.dx} = \\var{c}ln{x}\\]

\\[\\int{e^{\\var{d}x}}.dx = \\frac{e^{\\var{d}x}}{\\var{d}} \\]

Recombine and add the constant of integration

\\[ \\var{a}\\frac{x^\\var{b+1}}{\\var{b+1}} \\var{c}ln{x}+ \\frac{e^{\\var{d}x}}{\\var{d}} +C\\]

Use C as the constant on integration. Ensure variables and functions are in the numerator of any fractions.

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Integration by parts.

Integrate the following expression:

$f_{(x)}=\\frac{\\var{a}}{x^{\\var{b}}}\\var{c}{x}+e^{\\var{d}x}$

", "advice": "

\\[ f_{(x)}=\\frac{\\var{a}}{x^{\\var{b}}}\\var{c}{x}+e^{\\var{d}x} \\]

Each part of the expression can be integrated separately.

\\[\\int{\\frac{\\var{a}}{x^{\\var{b}}}}.dx=\\var{a}\\int{\\frac{1}{x^{\\var{b}}}}.dx=\\var{a}\\int{x^{-\\var{b}}}.dx =\\var{a}\\frac{x^\\var{-b+1}}{\\var{-b+1}}\\]

\\[ \\int{\\var{c}{x}=\\var{c}\\int{x}.dx }=\\var{c}\\frac{x^2}{2}\\]

\\[\\int{e^{\\var{d}x}}.dx = \\frac{e^{\\var{d}x}}{\\var{d}} \\]

Recombine and add the constant of integration

\\[ \\var{a}\\frac{x^\\var{-b+1}}{\\var{-b+1}} \\var{c}\\frac{x^2}{2}+ \\frac{e^{\\var{d}x}}{\\var{d}} +C\\]

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Use C as the constant on integration. Ensure variables and functions are in the numerator of any fractions.

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First order integral.

The following first order differential equation is given.

$\\frac{dy}{dx} = \\frac{\\var{a}}{{\\var{b}}x+{\\var{c}}}$, where $y=\\var{d}$ when $x=0$

", "advice": "

$\\frac{dy}{dx} = \\frac{\\var{a}}{{\\var{b}}x+{\\var{c}}}$, where $y=\\var{d}$ when $x=0$

\\[ \\int{\\frac{\\var{a}}{{\\var{b}}x+{\\var{c}}}dx} \\]

Move constant outside

\\[ \\var{a}\\int{\\frac{1}{{\\var{b}}x+{\\var{c}}}.dx} \\]

let \\[ u =\\var{b}x+\\var{c}\\]

\\[\\frac{du}{dx}=\\var{b} \\]

\\[\\therefore dx=\\frac{du}{\\var{b}}\\]

Sub back into equation

\\[ \\var{a}\\int{\\frac{1}{u}.\\frac{du}{\\var{b}}} \\]

The two x values cancel, leaving

\\[ \\frac{\\var{a}}{\\var{b}}\\int{\\frac{1}{u}.du}=\\frac{\\var{a}}{\\var{b}} ln(u)+C \\]

Sub back in the original denominator

\\[y= \\frac{\\var{a}}{\\var{b}} ln(\\var{b}x+\\var{c})+C \\]

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Find the general solution for the expression.

Enter values as integers or fractions.

$y=$[[0]]

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The general solution for part a can be expressed as,

$y=\\frac{\\var{a}}{\\var{b}}ln(\\var{b}x+\\var{c})+C$

Evaluate the constant of integration, C. Input values decimals.

$C=$[[0]]

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If the constant of integration can be expressed as

\\[ C =\\var{Cint} \\]

state the particular solution

$y=$[[0]]

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First order integral.

The following first order differential equation is given.

$\\frac{dy}{dx} = \\frac{\\var{a}x^\\var{b-1}}{{x^\\var{b}}+{\\var{c}}}$, where $y=\\var{d}$ when $x=0$

", "advice": "

$\\frac{dy}{dx} = \\frac{\\var{a}x^\\var{b-1}}{{x^\\var{b}}+{\\var{c}}}$, where $y=\\var{d}$ when $x=0$

\\[ \\int{\\frac{\\var{a}x^\\var{b-1}}{{x^\\var{b}}+{\\var{c}}}.dx} \\]

Move constant outside

\\[ \\var{a}\\int{\\frac{x^\\var{b-1}}{{x^\\var{b}}+{\\var{c}}}.dx} \\]

let \\[ u ={x^\\var{b}}+{\\var{c}}\\]

\\[\\frac{du}{dx}=\\var{b} x^\\var{b-1}\\]

\\[\\therefore dx=\\frac{du}{\\var{b} x^\\var{b-1}}\\]

Sub back into equation

\\[ \\var{a}\\int{\\frac{x^\\var{b-1}}{u}.\\frac{du}{\\var{b} x^\\var{b-1}}} \\]

The two x values cancel, leaving

\\[ \\frac{\\var{a}}{\\var{b}}\\int{\\frac{1}{u}.du}=\\frac{\\var{a}}{\\var{b}} ln(u)+C \\]

Sub back in the original denominator

\\[y= \\frac{\\var{a}}{\\var{b}} ln({x^\\var{b}}+{\\var{c}})+C \\]

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Find the general solution for the expression.

Enter values as decimals.

$y=$[[0]]

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The general solution for part a can be expressed as,

$\\frac{\\var{a}}{\\var{b}}ln(x^{\\var{b}}+\\var{c})+C$

Evaluate the constant of integration, C. Input values as decimals.

$C =$ [[0]]

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If the constant of integration is expressed as

\\[C = \\var{Cint} \\]

State the particular solution.

$y =$ [[0]]

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Separable equation for integration.

The following first order differential equation is given.

$\\frac{dy}{dx} = \\frac{x+\\var{a}}{\\var{b}y}$, where $y=\\var{c}$ when $x=0$

", "advice": "

$\\frac{dy}{dx} = \\frac{x+\\var{a}}{\\var{b}y}$, where $y=\\var{c}$ when $x=0$

Separate the variables

\\[ \\var{b}y.dy = (x+\\var{a}).dx \\]

Integrate each side

\\[ \\var{b}\\int{y.dy} = \\int{(x+\\var{a})}.dx \\]

\\[\\frac{\\var{b} y^2}{2}=\\frac{x^2}{2}+\\var{a}x+C \\] ...general solution

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Find the general solution for the expression.

$\\frac{dy}{dx} = \\frac{x+\\var{a}}{\\var{b}y}$

Enter values as integers or fractions. Use C as the constant of integration.

$y =$ [[0]]

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The general solution for part a can be expressed as,

\\[y =( {\\var{-a+1}}(\\frac{x^{2}}{2} + C))^{\\frac{1}{\\var{-a+1}}}\\]

Evaluate the constant of integration, C, for the initial conditions $y=\\var{b}$ when $x=0$. Input values as decimal.

$C=$ [[0]]

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For a constant of integration

\\[C=\\var{Cint}\\]

Sub this back into the general solution and write the particular solution in terms of y.

$y =$ [[0]]

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Separable equation for integration.

The following first order differential equation is given.

$\\frac{dy}{dx} = xy^\\var{a}$, where $y=\\var{b}$ when $x=0$

", "advice": "

$\\frac{dy}{dx} = xy^\\var{a}$, where $y=\\var{b}$ when $x=0$

Separate the variables

\\[ y^\\var{-a}.dy = x.dx \\]

Integrate each side

\\[ \\int{y^\\var{-a}.dy} = \\int{x}.dx \\]

\\[\\frac{y^\\var{-a+1}}{\\var{-a+1}}=\\frac{x^2}{2}+C \\] ...general solution

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Find the general solution for the expression.

$\\frac{dy}{dx} = xy^\\var{a}$

Enter values as integers or fractions. Use C as the constant of integration.

$y =$ [[0]]

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The general solution for part a can be expressed as,

\\[y =( {\\var{-a+1}}(\\frac{x^{2}}{2} + C))^{\\frac{1}{\\var{-a+1}}}\\]

Evaluate the constant of integration, C, for the initial conditions $y=\\var{b}$ when $x=0$. Input values as decimal.

$C=$ [[0]]

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For a constant of integration

\\[C=\\frac{\\var{b}^{\\var{-a+1}}}{\\var{-a+1}}=\\var{Cint}\\]

Sub this back into the general solution and write the particular solution in terms of y.

$y =$ [[0]]

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The 40 minutes for the exam have elapsed. Please end the exam now so you have time to print and upload your transcript to Brightspace.

"}, "timedwarning": {"action": "warn", "message": "

There are 5 minutes remaining in the exam.

"}}, "feedback": {"showactualmark": false, "showtotalmark": true, "showanswerstate": true, "allowrevealanswer": false, "advicethreshold": 0, "intro": "

This exam consists of

2 questions on integration by parts
1 separable equation
1 longer integral

There is a limit of 40 minutes on the exam. At the end of the exam, use the Print Result Summary to pdf. This will print your results summary and workings for each questions to a pdf. Upload this to Brightspace.

Your answers may still be correct even if marked incorrect by Numbas, these will all be reviewed in the transcripts in Brightspace to ensure all marks are allocated appropriately.

", "end_message": "

Thank you for completing the review week quiz.

Your answers may still be correct even if marked incorrect by Numbas, these will all be reviewed in the transcripts in Brightspace to ensure all marks are allocated appropriately.

Ensure you Print Result Summary to pdf and upload this to Brightspace: this is your submission, not a sheet of workings.

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