Super hoja - Resumen de formulas de calculo integral y diferencial PDF

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Resumen de formulas de calculo integral y diferencial...

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SUPER HOJA 𝑥 𝐿𝑛 ( ) = 𝐿𝑛𝑥 − 𝐿𝑛𝑦 𝑦

Potenc Potencia ia 𝑎−𝑛 =

1

𝐿𝑛𝑥 𝑛 = 𝑛𝐿𝑛𝑥

𝑎 = 𝑎𝑚−𝑛 𝑎𝑛 𝑚

𝐿𝑛(𝑒

𝑒

(𝑎𝑚 )𝑛 = 𝑎𝑚∗𝑛

(𝑎 ∗ 𝑏)𝑛 = 𝑎 𝑛 ∗ 𝑏 𝑛

𝑎 −𝑛 𝑏 ( ) =( ) 𝑏 𝑎

𝑛

√𝑎 = 𝑏 ⟺ 𝑏 = 𝑎 = 𝑛√𝑎𝑚

√𝑎 ∗ 𝑏 = √𝑎 ∗ √𝑏

𝑛

√ 𝑛√𝑎 =

𝑛

𝑎

𝑛

𝑛

𝑚

√𝑎

𝑚∗𝑛

√𝑎 𝑛

√ = 𝑛 𝑏 √𝑏

√𝑎𝑚 = ( √𝑎 )

𝑛

𝑛

𝑚

√𝑎𝑛 = |𝑎| = {

𝑛

𝑎 𝑠𝑖 𝑎 ≥ 0 −𝑎 𝑠𝑖 𝑎 < 0

Loga Logaritmac ritmac ritmación ión 𝐿𝑜𝑔𝑏 𝑁 ⟺ 𝑏 𝑛 = 𝑁

𝐿𝑜𝑔𝑏 (𝑥𝑦) = 𝐿𝑜𝑔𝑏 𝑥 + 𝐿𝑜𝑔𝑏 𝑦

𝑥 𝐿𝑜𝑔𝑏 ( ) = 𝐿𝑜𝑔𝑏 𝑥 − 𝐿𝑜𝑔𝑏 𝑦 𝑦

𝐿𝑜𝑔𝑏 𝑥 𝑛 = 𝑛𝐿𝑜𝑔𝑏 𝑥

𝐿𝑜𝑔𝑏 (𝑏 𝑛 ) = 𝑛

𝑏 𝐿𝑜𝑔𝑏(𝑛) = 𝑛

𝐿𝑛(𝑛)

=𝑛

sen 𝜃 =

=𝑛

𝐿𝑜𝑔𝑏 𝑁 =

𝐿𝑜𝑔 𝑁 = 𝑛 ⟺ 10𝑛 = 𝑁 𝐿𝑛 𝑁 = 𝑛 ⟺ 𝑒 𝑛 = 𝑁

𝑏 2 𝑏 2 ) +𝑐 −( ) 2 2

Trigon Trigonom om ometría etría

𝐿𝑜𝑔𝑁 𝐿𝑜𝑔𝑏

cos 𝜃 =

𝐿𝑛𝑁 𝐿𝑛𝑏

tan 𝜃 =

Polin Polinomi omi omio o

𝑛

𝑚 𝑎𝑛

𝑛)

𝐿𝑜𝑔𝑏 𝑁 =

Rad Radicación icación 𝑛

𝑥 2 + 𝑏𝑥 + 𝑐 = (𝑥 +

𝐿𝑛 (𝑥𝑦) = 𝐿𝑛𝑥 + 𝐿𝑛𝑦

𝑎𝑛

𝑎𝑚 ∗ 𝑎𝑛 = 𝑎𝑚+𝑛

𝑎 𝑛 𝑎𝑛 ( ) = 𝑛 𝑏 𝑏

Completar cuadrado

𝑎+𝑏+𝑐 𝑎 𝑏 𝑐 = + + 𝑑 𝑑 𝑑 𝑑 𝑟(𝑥) 𝑓(𝑥) = 𝑞(𝑥) + 𝑝(𝑥) 𝑝(𝑥)

𝑐𝑎𝑡𝑒𝑡𝑜 𝑜𝑝𝑢𝑒𝑠𝑡𝑜 𝑎 = 𝑐 ℎ𝑖𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑎

𝑐𝑎𝑡𝑒𝑡𝑜 𝑎𝑑𝑦𝑎𝑐𝑒𝑛𝑡𝑒 ℎ𝑖𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑎

=

𝑐𝑎𝑡𝑒𝑡𝑜 𝑎𝑑𝑦𝑎𝑐𝑒𝑛𝑡𝑒

𝑐𝑎𝑡𝑒𝑡𝑜 𝑜𝑝𝑢𝑒𝑠𝑡𝑜

=

𝑎

𝑐𝑎𝑡𝑒𝑡𝑜 𝑜𝑝𝑢𝑒𝑠𝑡𝑜

=

𝑏

=

𝑐

cot 𝜃 =

𝑐𝑎𝑡𝑒𝑡𝑜 𝑎𝑑𝑦𝑎𝑐𝑒𝑛𝑡𝑒

sec 𝜃 =

ℎ𝑖𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑎

𝑐𝑎𝑡𝑒𝑡𝑜 𝑎𝑑𝑦𝑎𝑐𝑒𝑛𝑡𝑒

𝑏 𝑐

𝑏 𝑎 𝑏

ℎ𝑖𝑝𝑜𝑡𝑒𝑛𝑢𝑠𝑎 𝑐 𝑐𝑠𝑐𝜃 = = 𝑐𝑎𝑡𝑒𝑡𝑜 𝑜𝑝𝑢𝑒𝑠𝑡𝑜 𝑎

(𝑥 + 𝑦)(𝑥 − 𝑦) = 𝑥 2 − 𝑦 2

Identid Identidad ad ades es tr trigo igo igonomé nomé nométrica trica tricass

(𝑥 − 𝑦)2 = 𝑥 2 − 2𝑥𝑦 + 𝑦 2

𝑠𝑒𝑛𝟐 𝜃 = 1 − 𝑐𝑜𝑠 2 𝜃

(𝑥 − 𝑦)3 = 𝑥 3 − 3𝑥 2𝑦 + 3𝑥𝑦2 − 𝑦 3

𝑠𝑒𝑐 2 𝜃 = 𝑡𝑎𝑛 2 𝜃 + 1

(𝑥 + 𝑦) = 𝑥 + 2𝑥𝑦 + 𝑦 2

2

2

𝑠𝑒𝑛𝟐 𝜃 + 𝑐𝑜𝑠2 𝜃 = 1

(𝑥 + 𝑦)3 = 𝑥 3 + 3𝑥 2𝑦 + 3𝑥𝑦 2 + 𝑦 3

𝑐𝑜𝑠2 𝜃 = 1 − 𝑠𝑒𝑛𝟐 𝜃

Factor Factorizac izac ización ión

𝑡𝑎𝑛2 𝜃 = sec2 𝜃 − 1

𝑥 − 𝑦 = (𝑥 + 𝑦 )(𝑥 − 𝑦 ) 2

2

𝑥 2 + 2𝑥𝑦 + 𝑦 2 = (𝑥 + 𝑦)2

𝑥 2 − 2𝑥𝑦 + 𝑦 2 = (𝑥 − 𝑦)2

𝑥 3 − 𝑦 3 = (𝑥 − 𝑦)(𝑥 2 + 𝑥𝑦 + 𝑦 2 )

𝑥 2 + 𝑏𝑥 + 𝑐 = (𝑥 + 𝑚)(𝑥 + 𝑛)

Donde m+n=b & m*n=c

𝑥 3 + 𝑦 3 = (𝑥 + 𝑦)(𝑥 2 − 𝑥𝑦 − 𝑦 2 )

𝑥 + 𝑦 = (𝑥 + 𝑦)(𝑥 −𝑥 𝑦+ 𝑛(𝑛−3) 𝑦 2 − ⋯ − 𝑥𝑦 𝑛−2 + 𝑦 𝑛−1 𝑛

𝑛

𝑛−1

𝑛−2

𝑥 − 𝑦 = (𝑥 − 𝑦)(𝑥 +𝑥 𝑦+ 𝑛(𝑛−3) 𝑦 2 + ⋯ + 𝑥𝑦 𝑛−2 + 𝑦 𝑛−1 𝑛

𝑛

𝑎𝑥 2 + 𝑏𝑥 + 𝑐 =

𝑛−1

𝑛−2

sec 2 𝜃 − tan2 𝜃 = 1 𝑐𝑠𝑐 2 𝜃 = 𝑐𝑜𝑡 2 𝜃 + 1

cot 2 𝜃 = csc 2 𝜃 − 1

csc 2 𝜃 − cot2 𝜃 = 1 𝑠𝑒𝑛𝜃 =

𝑐𝑠𝑐𝜃

1

; 𝑐𝑜𝑠𝜃 =

1 𝑠𝑒𝑐𝜃

𝑡𝑎𝑛𝜃 =

𝑐𝑜𝑡𝜃

1

; 𝑡𝑎𝑛𝜃 =

𝑠𝑒𝑛𝜃 𝑐𝑜𝑠𝜃

𝑐𝑜𝑡𝜃 =

𝑐𝑜𝑠𝜃 𝑠𝑒𝑛𝜃

−𝑏 ± √𝑏2 − 4𝑎𝑐 2𝑎

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Deriva Derivada da dass 𝑘=0 𝑢=1 𝑢 𝑛 = 𝑛𝑢 𝑛−1 ∗ 𝑢′ 𝑛𝑢 ′ 1 = − 𝑛+1 𝑢𝑛 𝑢′ 𝑢 √𝑢 = 2√𝑢 𝑢′ 𝑛 √𝑢 = 𝑛 𝑛−1 𝑛 √𝑢 𝑒𝑢 = 𝑢′𝑒𝑢 𝑎𝑢 = 𝑢 ′ 𝑎𝑢 𝐿𝑛𝑎 u′ Ln|u| = u 𝑢′ 𝐿𝑜𝑔𝑎 𝑢 = 𝐿𝑜𝑔𝑎 𝑒 𝑢 𝑠𝑒𝑛𝑢 = 𝑢 ′ 𝑐𝑜𝑠𝑢 𝑐𝑜𝑠𝑢 = −𝑢 ′ 𝑠𝑒𝑛𝑢 𝑡𝑎𝑛𝑢 = 𝑢 ′ sec 2 𝑢 𝑐𝑡𝑔𝑢 = −𝑢 ′ csc 2 𝑢 𝑠𝑒𝑐𝑢 = 𝑢 ′ 𝑠𝑒𝑐𝑢 ∗ 𝑡𝑎𝑛𝑢 𝑐𝑠𝑐𝑢 = −𝑢 ′ cscu ∗ 𝑐𝑡𝑔𝑢 𝑢′ 𝑎𝑟𝑐𝑠𝑒𝑛 𝑢 = 2 √1 − 𝑢 𝑢′ arccos 𝑢 = − √1 − 𝑢2 𝑢′ arctan 𝑢 = 1 + 𝑢2 𝑢 + 𝑣 − 𝑥 = 𝑢 ′ + 𝑣 ′ − 𝑥′ ′ 𝑢𝑥 = 𝑢 𝑥 + 𝑢𝑥 ′ 𝑢 𝑢 ′ 𝑥 − 𝑢𝑥 ′ = 𝑢2 𝑥 Inte Integrale grale graless ∫ 𝑑𝑥 = 𝑥 + 𝐶 ∫ 𝑘𝑑𝑥 = 𝑘𝑥 + 𝐶 ∫ 𝑥 𝑛 𝑑𝑥 = ∫

𝑥 𝑛+1 +𝐶 𝑛+1

𝑑𝑥 = 𝐿𝑛|𝑥| + 𝐶 𝑥

∫ 𝑎 𝑥 𝑑𝑥 =

𝑎𝑥 +𝐶 𝐿𝑛𝑎

∫ 𝑒 𝑥 𝑑𝑥 = 𝑒 𝑥 + 𝐶

∫ sen 𝑥 𝑑𝑥 = − cos 𝑥 + 𝐶 ∫ cos 𝑥 𝑑𝑥 = sen 𝑥 + 𝐶 ∫ tan 𝑥 𝑑𝑥 = −𝐿𝑛| cos 𝑥| + 𝐶

∫ 𝑐𝑡𝑔 𝑥 𝑑𝑥 = 𝐿𝑛|𝑠𝑒𝑛𝑥 | + 𝐶

∫ sec 𝑥 𝑑𝑥 = 𝐿𝑛|sec 𝑥 + tan 𝑥 | + 𝐶 ∫ csc 𝑥 𝑑𝑥 = 𝐿𝑛|csc 𝑥 − 𝑐𝑡𝑔𝑥 | + 𝐶

∫ sec 2 𝑥 𝑑𝑥 = tan 𝑥 + 𝐶

∫ csc 2 𝑥 𝑑𝑥 = − 𝑐𝑡𝑔 𝑥 + 𝐶 ∫ sec 𝑥 tan 𝑥 𝑑𝑥 = sec 𝑥 + 𝐶 ∫ csc 𝑥 cot 𝑥 𝑑𝑥 = − csc 𝑥 + 𝑐

𝑥 𝑑𝑥 = 𝑎𝑟𝑐 sen + 𝑐 𝑎 √𝑎2 − 𝑥 2 𝑑𝑥 1 𝑥 ∫ 2 𝑎𝑟𝑐 tan = +𝑐 𝑥 + 𝑎2 𝑎 𝑎 𝑑𝑥 1 𝑥−𝑎 |+ 𝑐 ∫ 2 ln | = 𝑥 − 𝑎 2 2𝑎 𝑥 + 𝑎 𝑑𝑥 1 𝑎+𝑥 ∫ 2 ln | = |+𝑐 2𝑎 𝑎 − 𝑥 𝑎 − 𝑥2 1 𝑥 𝑑𝑥 = 𝑎𝑟𝑐 sec + 𝑐 ∫ 𝑎 𝑎 𝑥√𝑥 2 − 𝑎 2 𝑑𝑥 2 ∫ √𝑥 = ln (𝑥 + ± 𝑎 2) + 𝐶 √𝑥 2+ 𝑎 2 𝟐 ∫

∫ √𝒂𝟐 − 𝒙𝟐 𝒅𝒙 =

𝒙 𝒂 𝒙 √𝒂𝟐 − 𝒙𝟐 + 𝒂𝒓𝒄 𝐬𝐞𝐜 + 𝒄 𝒂 𝟐 𝟐

𝒂𝟐 𝒙 ∫ √𝒙𝟐 +𝒂𝟐 𝒅𝒙 = √𝒙𝟐 +𝒂𝟐 + 𝐥𝐧 (𝒙 + √𝒙𝟐 +𝒂𝟐 ) + 𝑪 𝒂 𝟐

∫ 𝑎𝑟𝑐 sen 𝑥 𝑑𝑥 = 𝑥𝑎𝑟𝑐 sin 𝑥 + √1 − 𝑥 2 + 𝑐

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∫ 𝑎𝑟𝑐 cos 𝑥 𝑑𝑥 = 𝑥 𝑎𝑟𝑐 cos 𝑥 − √1 − 𝑥 2 ∫ 𝑎𝑟𝑐 tan 𝑥 𝑑𝑥 = 𝑥 𝑎𝑟𝑐 tan 𝑥 − ln √1 + 𝑥 2 + 𝑐 ∫ 𝑎𝑟𝑐 cot 𝑥 𝑑𝑥 = 𝑥 𝑎𝑟𝑐 cot 𝑥 + ln √1 + 𝑥 2 + 𝑐

1 1 ∫ 𝑠𝑒𝑛 2 𝑥 𝑑𝑥 = 𝑥 − sen 2𝑥 + 𝑐 4 2 1 1 ∫ 𝑐𝑜𝑠 2 𝑑𝑥 = 𝑥 + sen 2𝑥 + 𝑐 4 2 𝑐𝑜𝑠 𝑛+1 𝑥 +𝑐 ∫ 𝑐𝑜𝑠 𝑛 𝑥 sen 𝑥 𝑑𝑥 = − 𝑛+1 ∫ sen 𝑚𝑥 sen 𝑛𝑥 𝑑𝑥 = −

sen(𝑚 + 𝑛)𝑥 sen(𝑚 − 𝑛) + 2(𝑚 + 𝑛) 2(𝑚 − 𝑛)

cos(𝑚 + 𝑛)𝑥 cos(𝑚 − 𝑛) − ∫ sen 𝑚𝑥 cos 𝑛𝑥 𝑑𝑥 = − 2(𝑚 + 𝑛) 2(𝑚 − 𝑛)

∫ 𝑠𝑒𝑛 𝑛 𝑥 cos 𝑥 𝑑𝑥 =

𝑠𝑒𝑛𝑛+1 𝑥 +𝑐 𝑛+1

∫ 𝑥 sen 𝑥 𝑑𝑥 = sen 𝑥 − 𝑥 cos 𝑥 + 𝑐 ∫ 𝑥 cos 𝑥 𝑑𝑥 = cos 𝑥 + 𝑥 sen 𝑥 + 𝑐 𝑒 𝑎𝑥 +𝑐 𝑎 𝑎𝑥 𝑏 +𝑐 ∫ 𝑏 𝑎𝑥 𝑑𝑥 = 𝑎 ln 𝑏 𝑎𝑥 𝑒 ∫ 𝑥𝑒 𝑎𝑥 𝑑𝑥 = 2 (𝑎𝑥 − 1) + 𝑐 𝑎

∫ 𝑒 𝑎𝑥 𝑑𝑥 =

∫ 𝑥 𝑛 𝑒 𝑎𝑥 𝑑𝑥 = ∫

𝑥 𝑛 𝑒 𝑎𝑥 𝑛 − ∫ 𝑥 𝑛−1 𝑒 𝑎𝑥 𝑑𝑥 𝑎 𝑎

𝑎 ln 𝑏 𝑏𝑎𝑥𝑑𝑥 𝑏 𝑑𝑥 𝑏𝑎𝑥 + =− ∫ (𝑛 − 1)𝑥 𝑛−1 𝑛 − 1 𝑥 𝑛−1 𝑥𝑛 𝑎𝑥

∫ ln 𝑥 𝑑𝑥 = 𝑥 ln 𝑥 − 𝑥 + 𝑐 ∫ 𝑥 𝑛 ln 𝑥 𝑑𝑥 = 𝑥 𝑛−1 ⌈

ln 𝑥

𝑛+1

−

1 ⌉+𝑐 (𝑛 + 1)2

𝑒 𝑎𝑥 ln 𝑥 1 𝑒 𝑎𝑥 𝑑𝑥 − ∫ ∫ 𝑒 𝑎𝑥 ln 𝑥 𝑑𝑥 = 𝑥 𝑎 𝑎

∫

𝑑𝑥 = ln(ln 𝑥) + 𝑐 𝑥 ln 𝑥

𝑏

∫ 𝑓(𝑥)𝑑𝑥 = 𝐹(𝑥)⟧𝑏𝑎 = 𝐹(𝑏) − 𝐹(𝑎) 𝑎...