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a number of important techniques:indefinite integrals of more complicated functions

• how to change unfamiliar integrals• find in a table,• evaluate with a computer.

We often have to rewrite an integral to match it to a

standard formula.

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Recall

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how to express a rational function (a quotient of polynomials) as a sumof simpler fractions, called partial fractions, which are easily integrated

For example

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𝒚′ = 2 − 6𝑥 𝑑𝑥 →𝒚′ = 2𝑥 − 3𝑥2 + 𝑐

𝑦′ 0 = 4 → 4 = 2 ∙ 0 − 3 ∙ 02 + 𝑐 → 𝑐 = 4

𝑦′ = 2𝑥 − 3𝑥2 + 4

𝒅𝟐𝒚

𝒅𝒙𝟐= 𝟐 − 𝟔𝒙 , 𝒚′ 𝟎 = 𝟒, 𝒚 𝟎 = 𝟏

Solve the initial value problem

𝑦 = 2𝑥 − 3𝑥2 + 4 𝑑𝑥 → 𝑦 = 𝑥2 − 𝑥3 + 4𝑥 + 𝑐

𝑦 0 = 1 → 1 = 02 − 03 + 4 ∙ 0 + 𝑐 → 𝑐 = 1

𝑦 = 𝑥2 − 𝑥3 + 4𝑥 + 1

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Washer method

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Solve the initial value problem explicitly 𝒅𝒚

𝒅𝒙= −

𝟏

𝒙𝟐−𝟑

𝒙𝟒+ 𝟏𝟐 and y=3

when x=1Evaluate the derivative of

following function.

𝒚 = 𝒙𝒙𝟏𝟎

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