72:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Why does every indefinite integral end with ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

+ C

"}}],"?"]}]]}] 73:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Antiderivatives of the same function differ by a constant."}]]}] ad:T4e2,

\displaystyle\int x^n\,dx

ae:T4a4,

n \ne -1

74:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$ad"}}]," for ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$ae"}}],"?"]}]]}] af:T941,

\dfrac{x^{n+1}}{n+1} + C

75:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$af"}}]}]]}] b0:T64a,

\displaystyle\int \dfrac{dx}{x}

76:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b0"}}],"?"]}]]}] 77:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

\\log|x| + C

"}}]," (keep the modulus)."]}]]}] b1:T4e2,

\displaystyle\int a^x\,dx

78:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b1"}}],"?"]}]]}] b2:T895,

\dfrac{a^x}{\log a} + C

79:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b2"}}]}]]}] b3:T5d2,

y = x^2 + C

7a:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Picture: parabolas ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b3"}}]," stacked vertically. What do they share?"]}]]}] 7b:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["The same slope ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

2x

"}}]," at every ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

x

"}}],": they are all antiderivatives of ",["$","span","code-2",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

2x

"}}],"."]}]]}] 7c:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"When do you use substitution?"}]]}] 7d:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"When a function and its derivative appear together in the integrand."}]]}] 7e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"

\\displaystyle\\int \\tan x\\,dx

"}}],"?"]}]]}] b4:T4b3,

\log|\sec x| + C

7f:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b4"}}]}]]}] 80:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"

\\displaystyle\\int \\sec x\\,dx

"}}],"?"]}]]}] b5:T662,

\log|\sec x + \tan x| + C

81:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b5"}}]}]]}] b6:Ta34,

\displaystyle\int \dfrac{f'(x)}{f(x)}\,dx

82:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b6"}}],"?"]}]]}] b7:T4c5,

\log|f(x)| + C

83:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$b7"}}]}]]}] b8:T558,

\displaystyle\int \sin^2 x\,dx

84:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["First step for ",["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b8"}}],"?"]}]]}] b9:T9a1,

\sin^2 x = \dfrac{1 - \cos 2x}{2}

85:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Use ",["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$b9"}}],"."]}]]}] ba:T6dc,

\displaystyle\int \cos x\,e^{\sin x}\,dx

86:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$ba"}}],"?"]}]]}] bb:T583,

e^{\sin x} + C

87:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$bb"}}]," (put ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

t = \\sin x

"}}],")."]}]]}] bc:Ta2f,

\displaystyle\int \dfrac{dx}{x^2 + a^2}

88:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$bc"}}],"?"]}]]}] bd:Tc7f,

\dfrac1a\tan^{-1}\dfrac xa + C

89:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$bd"}}]}]]}] be:Ta33,

\displaystyle\int \dfrac{dx}{x^2 - a^2}

8a:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$be"}}],"?"]}]]}] bf:T12cc,

\dfrac{1}{2a}\log\left|\dfrac{x - a}{x + a}\right| + C

8b:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$bf"}}]}]]}] c0:Tf0d,

\displaystyle\int \dfrac{dx}{\sqrt{a^2 - x^2}}

8c:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$c0"}}],"?"]}]]}] c1:T8b9,

\sin^{-1}\dfrac xa + C

8d:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$c1"}}]}]]}] c2:Tf09,

\displaystyle\int \dfrac{dx}{\sqrt{x^2 + a^2}}

8e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$c2"}}],"?"]}]]}] c3:T1326,

\log\left|x + \sqrt{x^2 + a^2}\right| + C

8f:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$c3"}}]}]]}] 90:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Quadratic denominator that does not factorise: first step?"}]]}] 91:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Complete the square."}]]}] c4:T9ce,

\dfrac{px + q}{ax^2 + bx + c}

92:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["How do you split ",["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$c4"}}],"?"]}]]}] c5:T7ee,

px + q = A(2ax + b) + B

93:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$c5"}}],": a log part and a special-form part."]}]]}] 94:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Proper rational function?"}]]}] 95:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Numerator degree less than denominator degree."}]]}] c6:T6bb,

(x - a)^2(x - b)

96:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Partial fractions for ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$c6"}}],"?"]}]]}] c7:T133e,

\dfrac{A}{x - a} + \dfrac{B}{(x - a)^2} + \dfrac{C}{x - b}

97:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$c7"}}]}]]}] c8:T677,

\dfrac{1}{x(x + 1)}

98:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Split ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$c8"}}],"."]}]]}] c9:Ta77,

\dfrac1x - \dfrac{1}{x + 1}

99:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$c9"}}]}]]}] 9a:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Improper rational function: what first?"}]]}] 9b:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Divide, then split the proper remainder."}]]}] 9c:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Integration by parts formula?"}]]}] ca:T7af,

\displaystyle\int u\,dv = uv - \int v\,du

9d:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$ca"}}]}]]}] 9e:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Order for choosing ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

u

"}}]," in by parts?"]}]]}] 9f:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Inverse trig, log, algebraic, trig, exponential."}]]}] cb:Ta06,

\displaystyle\int e^x[f(x) + f'(x)]\,dx

a0:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$cb"}}],"?"]}]]}] cc:T568,

e^xf(x) + C

a1:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$cc"}}]}]]}] a2:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"

\\displaystyle\\int \\log x\\,dx

"}}],"?"]}]]}] cd:T540,

x\log x - x + C

a3:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$cd"}}]," (by parts with ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

u = \\log x

"}}],")."]}]]}] a4:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Fundamental theorem of calculus?"}]]}] ce:T4e9,

F' = f

cf:Tb1e,

\displaystyle\int_a^b f(x)\,dx = F(b) - F(a)

a5:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["If ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"$ce"}}]," on ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

[a, b]

"}}],", ",["$","span","code-2",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$cf"}}],"."]}]]}] d0:T7a8,

\displaystyle\int_{-a}^{a} f(x)\,dx

a6:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":[["$","span","code-0",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$d0"}}]," when ",["$","span","code-1",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

f

"}}]," is odd? When even?"]}]]}] d1:T77a,

2\displaystyle\int_0^a f(x)\,dx

a7:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":["Odd: ",["$","span","code-0",{"className":"SmxAaG_mathInline","data-math":"inline","tabIndex":"$undefined","dangerouslySetInnerHTML":{"__html":"

0

"}}],". Even: ",["$","span","code-1",{"className":"SmxAaG_mathInline SmxAaG_mathInlineScroll","data-math":"inline","tabIndex":0,"dangerouslySetInnerHTML":{"__html":"$d1"}}],"."]}]]}] a8:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Substitution in a definite integral: what else changes?"}]]}] a9:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"The limits, which become values of the new variable."}]]}] aa:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"Which part of definite integrals is removed from the 2026–27 syllabus?"}]]}] ab:["$","div",null,{"className":"SmxAaG_md","lang":"$undefined","children":[["$","p","p-0",{"children":"The definite integral as the limit of a sum."}]]}]

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