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Basic indefinite integrals, Basic definite integrals, integration by substitution

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rebel

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rebelmaths

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$f(x) = \\var{a}x - \\var{b}$

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$f(x) = \\frac{1}{\\var{c}}+ \\frac{2}{\\var{d}}x^2 - \\frac{3}{\\var{f}}x^3$

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$f(x) = e^\\var{g}$

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$f(x) = e^{\\var{g}x}$

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Antiderivatives

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rebel

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rebelmaths

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Find the most general antiderivatives of the functions. Use the letter C to represent an unknown constant.

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Don't forget to include the unknown constant C.

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Definite Integrals

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rebel

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rebelmaths

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Rebelmaths

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$\\int_\\var{a}^\\var{b}(1 + \\var{c}x)\\mathrm{dx}$

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$\\int_\\var{d}^\\var{f} (x^2 + \\var{g}x-\\var{h})\\mathrm{dx}$

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$\\int_\\var{j}^\\var{k}(x^3-\\var{l}x^2)\\mathrm{dx}$

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Evaluate the integrals:

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First integrate the function and then substitute in the given limits.

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Indefinite Integrals

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rebel

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rebelmaths

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Solve the following indefinite integrals, using $C$ to represent an unknown constant.

Indefinite Integrals

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$\\int(x^4-\\var{a}x^3+\\var{b}x-\\var{c})\\mathrm{dx}$

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$\\int(u+\\var{d})(2u+\\var{f})\\mathrm{du}$

", "answer": "2u^3/3+u^2({f}+2{d})/2+{d}{f}u+C", "showPreview": true, "checkingType": "absdiff", "checkingAccuracy": 0.001, "failureRate": 1, "vsetRangePoints": 5, "vsetRange": [0, 1], "checkVariableNames": false, "singleLetterVariables": false, "allowUnknownFunctions": true, "implicitFunctionComposition": false, "valuegenerators": [{"name": "c", "value": ""}, {"name": "u", "value": ""}]}], "partsMode": "all", "maxMarks": 0, "objectives": [], "penalties": [], "objectiveVisibility": "always", "penaltyVisibility": "always"}, {"name": "cormac's copy of Definite Integrals 2", "extensions": [], "custom_part_types": [], "resources": [], "navigation": {"allowregen": true, "showfrontpage": false, "preventleave": false, "typeendtoleave": false}, "contributors": [{"name": "cormac breen", "profile_url": "https://numbas.mathcentre.ac.uk/accounts/profile/306/"}], "parts": [{"vsetrange": [0, 1], "checkingaccuracy": 0.001, "showCorrectAnswer": true, "checkvariablenames": false, "type": "jme", "marks": 1, "variableReplacementStrategy": "originalfirst", "expectedvariablenames": [], "variableReplacements": [], "scripts": {}, "vsetrangepoints": 5, "showFeedbackIcon": true, "showpreview": true, "answer": "e^2/(e+1)", "prompt": "

$\\int_0^1(x^e+e^x)\\mathrm{dx}$

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$\\int_1^\\var{b}(\\frac{x^3+\\var{c}x^6}{x^4})\\mathrm{dx}$

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You may have $\\ln$ terms in your answer.

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$\\int_0^\\var{d}\\sqrt{\\frac{3}{z}}\\mathrm{dz}$

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Definite Intgerals

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rebel

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rebelmaths

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Rebelmaths

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First integrate the function and then substitute in the limits given

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Find the following definite integrals

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a)
\$I=\\int_1^\\var{b1}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\$
Expand the parentheses to obtain:

\n

\$\\begin{eqnarray*}I &=& \\int_1^\\var{b1} \\simplify[std]{{a1 ^ 2} * x ^ 4 + {2 * a1 * c1} * x ^ 3+ {c1 ^ 2+2*a1*d1} * x ^ 2 + {2 * c1 * d1} * x+ {d1 ^ 2} }\\;dx\\\\ &=&\\left[\\simplify[std]{{a1 ^ 2}/5 * x ^ 5 + {2 * a1 * c1}/4 * x ^ 4+ {c1 ^ 2+2*a1*d1}/3 * x ^ 3 + {2 * c1 * d1}/2 * x^2+ {d1 ^ 2}x }\\right]_1^\\var{b1}\\\\ &=&\\var{tans1}\\\\ \\\\&=&\\var{ans1}\\mbox{ to 2 decimal places} \\end{eqnarray*} \$

\n

b)
\$\\begin{eqnarray*}I&=&\\int_0^{\\var{b2}}\\simplify[std]{1/(x+{m2})}\\;dx\\\\ &=&\\left[\\ln(x+\\var{m2})\\right]_0^{\\var{b2}}\\\\ &=& \\ln(\\var{b2+m2})-\\ln(\\var{m2})\\\\ &=&\\ln\\left(\\frac{\\var{b2+m2}}{\\var{m2}}\\right)\\\\ &=&\\var{ans2}\\mbox{ to 2 decimal places} \\end{eqnarray*} \$

\n

c)
\$I=\\int_0^\\pi\\simplify[std]{x * ({w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x))}\\;dx\$
We use integration by parts.

\n

Recall that:
\$\\int u\\frac{dv}{dx}\\;dx=uv-\\int \\frac{du}{dx}\\;v\\;dx\$
Here we set $u=x$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{ {w} * Sin({m3} * x) + {1 -w} * Cos({m3} * x)}$

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Hence \$v=\\simplify[std]{({-w}/ {m3}) * Cos({m3} * x) + {1 -w} * (({1-w}/ {m3}) * Sin({m3} * x))}\$

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So \$\\begin{eqnarray*} I&=&\\left[\\simplify[std]{{-w}*((x / {m3}) * Cos({m3} * x)) + {1 -w} * ((x / {m3}) * Sin({m3} * x))}\\right]_0^\\pi -\\int_0^\\pi\\simplify[std]{ ({ -w} / {m3} )* Cos({m3} * x) + {1 -w} * (1 / {m3} * Sin({m3} * x))}\\;dx\\\\ &=&\\simplify[std]{({-w*cos(m3*pi)})*({pi}/{m3})}-\\left[\\simplify[std]{{ -w} * (1 / {m3 ^ 2})* Sin({m3} * x) -({1 -w} * (1 / {m3 ^ 2}) * Cos({m3} * x))}\\right]_0^\\pi\\\\ &=& \\var{ans3}\\mbox{ to 2 decimal places} \\end{eqnarray*} \$
d)

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\$I=\\int_0^{\\var{b4}}\\simplify[std]{x ^ {m4} * Exp({n4} * x)}\\;dx\$

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Use integration by parts twice with $u=x^2$ and $\\displaystyle \\frac{dv}{dx}=\\simplify[std]{e^({n4}x)}\\Rightarrow v = \\simplify[std]{1/{n4}e^({n4}x)}$
\$\\begin{eqnarray*} I&=&\\left[\\simplify[std]{x^2/{n4}Exp({n4} * x)}\\right]_0^{\\var{b4}}+\\simplify[std]{2/{abs(n4)}DefInt(x*Exp({n4} * x),x,0,{b4})}\\\\ &=&\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\left[\\simplify[std]{x/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}+\\simplify[std]{1/{abs(n4)}DefInt(e^({n4}x),x,0,{b4})}\\right)\\\\ &=&\\simplify[std]{{b4^2}/{n4}*e^{p}-2/{n4}}\\left(\\simplify[std]{{b4}/{n4}*e^{p}-1/{n4}}\\left[\\simplify[std]{1/{n4}*e^({n4}*x)}\\right]_0^{\\var{b4}}\\right)\\\\ &=&\\simplify[std]{({b4 ^ 2} / {n4}) * Exp({p}) -(({2 * b4} / {n4 ^ 2}) * Exp({p})) + (2 / {n4 ^ 3}) * (Exp({p}) -1)}\\\\ &=&\\var{ans4}\\mbox{ to 4 decimal places} \\end{eqnarray*} \$

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\$I=\\int_1^{\\var{b1}}\\simplify[std]{({a1} * x ^ 2 + {c1} * x + {d1})^2}\\;dx\$

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$I=\\;\\;$[[0]]

\n

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Evaluate $\\int_1^{\\,m}(ax ^ 2 + b x + c)^2\\;dx$, $\\int_0^{p}\\frac{1}{x+d}\\;dx,\\;\\int_0^\\pi x \\sin(qx) \\;dx$, $\\int_0^{r}x ^ {2}e^{t x}\\;dx$

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rebelmaths

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Evaluate the following definite integral.

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5 indefinite integrals containing exponential functions

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rebelmaths

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Integrate $f(x)=e^{\\var{a1}x}$ with respect to $x$.

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Integrate $f(x)=\\var{a1}e^{\\var{a2}x}$ with respect to $x$.

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Integrate $f(x)=\\var{a3}\\exp(\\var{a4}x)+\\var{a1}\\exp(\\var{a5}x)$ with respect to $x$.

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Integrate $f(x)=\\dfrac{2}{\\var{a5}}\\exp(\\var{a1}x)+\\dfrac{1}{\\var{a3}}\\exp(\\var{a4}x)$ with respect to $x$.

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The basic results are

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$\\int(e^{x})dx=e^{x}+c$ or $\\int\\exp({x})dx=\\exp({x})+c$

\n

$\\int(e^{kx})dx=\\dfrac{1}{k}e^{kx}+c$ or $\\int\\exp({kx})dx=\\dfrac{1}{k}\\exp({kx})+c$

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Don't forget the constant!

", "statement": "

The basic results are

\n

$\\int(e^{x})dx=e^{x}+c$ or $\\int\\exp({x})dx=\\exp({x})+c$

\n

$\\int(e^{kx})dx=\\dfrac{1}{k}e^{kx}+c$ or $\\int\\exp({kx})dx=\\dfrac{1}{k}\\exp({kx})+c$

\n

Don't forget the constant!

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Using
\$\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\$ for any number $n \\neq -1$ we have
\$\\begin{eqnarray*}\n\t \n\t \\simplify[std]{Int({c}*x^{m}+{d}*x ^ ({b} / {n}),x)} &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} +{d}* x ^ ({b} / {n} + 1) / ({b} / {n} + 1) + C }\\\\\n\t \n\t &=&\\simplify[std]{ ({c} / {m + 1}) * x ^ {m + 1} + ({d*n} / {b + n}) * x ^ ({b + n} / {n}) + C}\n\t \n\t \\end{eqnarray*}\$

\n\t \n\t \n\t \n\t", "rulesets": {"std": ["all", "!collectNumbers", "fractionNumbers"], "surdf": [{"pattern": "a/sqrt(b)", "result": "(sqrt(b)*a)/b"}]}, "parts": [{"stepsPenalty": 0, "prompt": "\n\t\t\t

$\\simplify[std]{f(x) = {c}x ^ {m} + {d}*x^({b}/{n})}$

\n\t\t\t

$\\displaystyle \\int\\;f(x)\\,dx=\\;$[[0]]

\n\t\t\t

Input all numbers as integers or fractions and not decimals. Remember to include the constant of integration $C$.

\n\t\t\t

Click on Show steps to get more information. You will not lose any marks by doing so.

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Input all numbers as integers or fractions and not decimals.

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The indefinite integral of a power $x^n$ where $n\\neq -1$ is \$\\int \\;x^n\\;dx=\\frac{x^{n+1}}{n+1}+C\$

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Integrate the following function $f(x)$.

\n\t

Input the constant of integration as $C$.

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\n

indefinite integration

\n

Find $\\displaystyle \\int ax ^ m+ bx^{c/n}\\;dx$.

\n

rebel

\n

rebelmaths

\n

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