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Integration

Course Material

Gyeongsang National University

Dept. of Information & Communication Engineering

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Basic Integration

• Integration

– Integration is a reverse process of the differentiation – We can derive 𝑓(𝑥) from the derivative of a function

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Basic Integration

• Integration

– Example a) sin(3𝑥 + 2) b) 1 𝑥2 c) 𝑒−3𝑧

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Linearity of Integration

• Integration is a linear operator

– Integration of (𝑓 + 𝑔) = Integration of 𝑓+ Integration of 𝑔

– Integration of (𝑘𝑓) = 𝑘 x Integration of 𝑓

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Definite Integral

• Definite integral

– Indefinite integral includes an integral constant 𝑐 – Integration with lower and upper limits

– ׬₁4𝑥2 𝑑𝑥 = 𝑥3

3 + 𝑐 ₁ 4

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Definite Integral

• Definite integral – area of a function

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Definite Integral

• Definite integral – area of a function

– Area between lower 𝑥 = 𝑎 and upper 𝑥 = 𝑏 limits

– Example a) ׬₁2𝑥2 + 1𝑑𝑥 b) ׬₂1𝑥2 + 1𝑑𝑥 c) ׬₀𝜋sin 𝑥 𝑑𝑥 න 𝑎 𝑏 𝑦 𝑑𝑥 = 𝐴 𝑏 − 𝐴(𝑎)

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Integration by Parts

• Integration by parts

– Derive from a product rule

– Integration by parts of definite integral

𝑑

𝑑𝑥

(𝑢𝑣) =

𝑑𝑢

𝑑𝑥

𝑣+𝑢

𝑑𝑣

𝑑𝑥

න 𝑢

𝑑𝑣

𝑑𝑥

𝑑𝑥 = 𝑢𝑣 − න 𝑣

𝑑𝑢

𝑑𝑥

𝑏

_𝑑𝑣

𝑏

_𝑑𝑢

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Integration by Parts

• Integration by parts

– ׬₀2𝑥𝑒𝑥 𝑑𝑥 – ׬₀2𝑥2𝑒𝑥 𝑑𝑥 – ׬ 𝑒𝑡 _{sin 𝑡 𝑑𝑡} 𝑢 = 𝑥, 𝑑𝑣 𝑑𝑥 = 𝑒 𝑥 න 𝑎 𝑏 𝑢𝑑𝑣 𝑑𝑥 𝑑𝑥 = 𝑢𝑣 𝑎 𝑏 _{− න} 𝑎 𝑏 𝑣𝑑𝑢 𝑑𝑥 𝑑𝑥

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Integration by Substitution

• Integration by substitution

– Integration by substituting one variable into another variable – ׬(3𝑥 + 1)2.7 _𝑑𝑥 – ׬_π2πsin 𝑡 cos2𝑡 𝑑𝑡 𝑧 = 3𝑥 + 1 𝑑𝑧 = 3𝑑𝑥 𝑧 = cos 𝑡 𝑑𝑧 = −sin𝑡𝑑𝑡 න cosπ cos2π sin 𝑡 𝑧2 1 −sin𝑡𝑑𝑧

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Integration by Substitution

• Integration by substitution

– Example a) ׬ 4 5𝑥−7𝑑𝑥 b) ׬_𝑡₂𝑡₊₁𝑑𝑡 c) ׬_𝑒_𝑡/2𝑒𝑡/2₊₁𝑑𝑡

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Integration by Partial Fraction

• Integration by Partial Fraction

– Integration by decomposing a function into partial fractions

– Integral that seems not to be integrated can be integrated by partial fractions.

– ׬ 1

𝑥3+𝑥 𝑑𝑥

Partial fractional decomposition

1 𝑥3 _{+ 𝑥} = 1 𝑥(𝑥2 _{+ 1)} = 𝐴 𝑥 + 𝐵𝑥 + 𝐶 𝑥2 _{+ 1} = 𝐴 𝑥2 + 1 + 𝑥(𝐵𝑥 + 𝐶) 𝑥(𝑥2 _{+ 1)} 𝐴 𝑥2 + 1 + 𝑥 𝐵𝑥 + 𝐶 = 𝐴 + 𝐵 𝑥2 + 𝐶𝑥 + 𝐴 = 1

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Integration by Partial Fraction

• Integration by Partial Fraction

– Example a) ׬ 𝑥+3

𝑥2+𝑥 𝑑𝑥

b) ׬ 13𝑥−4

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