Integral Cheat Sheet - © 2005 paul dawkins integrals definitions definite integral: ′= −∫ ′ ∫integral of a constant: ( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out: Integral substitution \int f\left(g\left(x\right)\right)\cdot g^'\left(x\right)dx=\int f\left(u\right)du,\:\quad u=g\left(x\right) definite integrals rules definite. ⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule: (𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥
( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out: (𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥 Integral substitution \int f\left(g\left(x\right)\right)\cdot g^'\left(x\right)dx=\int f\left(u\right)du,\:\quad u=g\left(x\right) definite integrals rules definite. ′= −∫ ′ ∫integral of a constant: ⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule: © 2005 paul dawkins integrals definitions definite integral:
(𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥 ( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out: ⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule: Integral substitution \int f\left(g\left(x\right)\right)\cdot g^'\left(x\right)dx=\int f\left(u\right)du,\:\quad u=g\left(x\right) definite integrals rules definite. © 2005 paul dawkins integrals definitions definite integral: ′= −∫ ′ ∫integral of a constant:
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(𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥 Integral substitution \int f\left(g\left(x\right)\right)\cdot g^'\left(x\right)dx=\int f\left(u\right)du,\:\quad u=g\left(x\right) definite integrals rules definite. © 2005 paul dawkins integrals definitions definite integral: ′= −∫ ′ ∫integral of a constant: ⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule:
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′= −∫ ′ ∫integral of a constant: (𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥 ⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule: © 2005 paul dawkins integrals definitions definite integral: ( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out:
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(𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥 ′= −∫ ′ ∫integral of a constant: Integral substitution \int f\left(g\left(x\right)\right)\cdot g^'\left(x\right)dx=\int f\left(u\right)du,\:\quad u=g\left(x\right) definite integrals rules definite. ( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out: ⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule:
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⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule: Integral substitution \int f\left(g\left(x\right)\right)\cdot g^'\left(x\right)dx=\int f\left(u\right)du,\:\quad u=g\left(x\right) definite integrals rules definite. ′= −∫ ′ ∫integral of a constant: ( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out: © 2005 paul dawkins integrals definitions definite integral:
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′= −∫ ′ ∫integral of a constant: (𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥 ( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out: Integral substitution \int f\left(g\left(x\right)\right)\cdot g^'\left(x\right)dx=\int f\left(u\right)du,\:\quad u=g\left(x\right) definite integrals rules definite. © 2005 paul dawkins integrals definitions definite integral:
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′= −∫ ′ ∫integral of a constant: © 2005 paul dawkins integrals definitions definite integral: (𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥 ⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule: ( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out:
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(𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥 Integral substitution \int f\left(g\left(x\right)\right)\cdot g^'\left(x\right)dx=\int f\left(u\right)du,\:\quad u=g\left(x\right) definite integrals rules definite. ⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule: ′= −∫ ′ ∫integral of a constant: ( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out:
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(𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥 ′= −∫ ′ ∫integral of a constant: ( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out: © 2005 paul dawkins integrals definitions definite integral: ⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule:
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′= −∫ ′ ∫integral of a constant: © 2005 paul dawkins integrals definitions definite integral: ⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule: ( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out: (𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥
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⋅ (𝑥 ) 𝑥= ⋅∫ 𝑥 𝑥 ∫sum/difference rule: Integral substitution \int f\left(g\left(x\right)\right)\cdot g^'\left(x\right)dx=\int f\left(u\right)du,\:\quad u=g\left(x\right) definite integrals rules definite. © 2005 paul dawkins integrals definitions definite integral: ( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out: ′= −∫ ′ ∫integral of a constant:
Integral Substitution \Int F\Left(G\Left(X\Right)\Right)\Cdot G^'\Left(X\Right)Dx=\Int F\Left(U\Right)Du,\:\Quad U=G\Left(X\Right) Definite Integrals Rules Definite.
( ) 𝑥=𝑥⋅ ( ) ∫taking a constant out: ′= −∫ ′ ∫integral of a constant: © 2005 paul dawkins integrals definitions definite integral: (𝑥 )± (𝑥 𝑥=∫ (𝑥) 𝑥±∫ 𝑥) 𝑥