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27 - Strategy

# 27 - Strategy - Strategies for Integration Know the basic...

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Strategies for Integration Know the basic formulas! If an integrand is not a basic formula, then 1. Simplify if possible Ex. e x e 2 + x ( ) ! dx Ex. x 2 + 4 ( ) ! 2 dx Ex. 1 + tan 2 x ! dx Ex. 1 x 2 + 4 x + 5 ! dx

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2. Use u-substitution Ex. x cos x 2 ( ) ! e sin x 2 ( ) dx 3. For rational functions A. Divide if necessary Ex. x 2 + 1 x + 1 ! dx B. Use partial fractions 4. Use integration by parts – look for a product in the integrand or a term that we cannot integrate but can take the derivative of.

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5. If you have a radical a 2 ± u 2 and it isn’t a basic formula, use trig substitution. (Remember 1 a 2 ! x 2 dx = sin ! 1 x a " # \$ % & ( + C ) 6. If you have trigonometric integrals, use the identities below or change to sine and cosine. Even powers only: sin 2 x = 1 2 ! 1 2 cos2 x cos 2 x = 1 2 + 1 2 cos2 x It there is an odd power: sin 2 x = 1 ! cos 2 x cos 2 x = 1 ! sin 2 x Ex. 2sin x x sec x ! dx Ex. 1 5 + x 2 ! dx
Ex. 1 5 + x 2 ! dx Ex. Write the integral 3 x ! 2 ( ) sin ! 1 x " dx in the form sin ! 1 ( x ) P ( x ) + 1 ! x Q ( x ) with P and Q polynomials. Find the constant term in P.

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Ex.
Do: Evaluate the following integrals. 1. e t 1 + e t dt ! 2. sec 3 4 x ( ) sin 5 4 x ( ) dx !

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Integral Review I . Integrate the following. Do not

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27 - Strategy - Strategies for Integration Know the basic...

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