VIII - 南華大學



VIII. PRINCIPLES OF INTEGRAL EVALUATION

8.1 Integration by Parts

1. Formula for integration by parts: [pic], or [pic] for [pic], [pic], [pic], [pic]

Key points for using integration by parts: (a) v(x) is easy to be derived. (b) The evaluation of [pic] is easier than [pic].

2. Integration by parts for definite integrals: [pic]

[pic]

[pic]

[pic]

[pic]

[pic]

8.2 Trigonometric integrals

1. Integrating powers of sine and cosine: [pic] [pic]

2. Integrating products of sines and cosines:[pic], m, n are positive integers.

(a) If m and n are even: Use the identities [pic] and [pic] to reduce the powers on sinx and cosx.

(b) If m is odd: Split off a factor of sinx. Applying the identity [pic]. Make the substitution [pic].

(c) If n is odd: Split off a factor of cosx. Applying the identity [pic]. Make the substitution [pic].

3. Integrating powers of tangent and secant: [pic] [pic]

4. Integrating products of tangent and secants:[pic], m and n are positive integers.

(a) If n is even: Split off a factor of [pic]. Apply the identity [pic]. Make the substitution u=tanx.

(b) If m is odd: Split off a factor of secxtanx. Apply the identity [pic]. Make the substitution u=secx.

(c) If m is even and n is odd: Use the identities [pic] to reduce the integrand to powers of secx alone. Then use the reduction formula for powers of secx.

8.3 Trigonometric Substitutions

1. The method of trigonometric substitution

|Expression in the |Substitution |Restriction on [pic] |Simplification |

|integrand | | | |

|[pic] |[pic] |[pic] |[pic] |

|[pic] |[pic] |[pic] |[pic] |

|[pic] |[pic] |[pic], if [pic] |[pic] |

| | |[pic], if [pic] | |

2. Integrals involving [pic]

8.4 Integrating Rational Functions by Partial Fractions

1. Linear factor rule: For each factor of the form [pic], the partial fraction decomposition contains the following sum of m partial fractions: [pic] where [pic] are constant to be determined.

2. Quadratic factor rule: For each factor of the form [pic], the partial fraction decomposition contains the following sum of m partial fractions: [pic] where [pic], [pic] are constant to be determined.

3. Improper rational function can be integrated by performing a long division and expressing the function as the quotient plus the remainder over the divisor.

[pic]

[pic]

[pic]

8.5 Using Table of Integrals and Computer Algebra System

1. Matches requiring substitutions

2. Matches requiring reduction formulas

3. Matches requiring special substitutions

[pic]

[pic]

[pic]

[pic]

8.6 Improper Integrals

1. Improper integrals: the integrals with infinite intervals of integration or infinite discontinuities within the interval of integration

2. Integrals over infinite intervals

(a) The improper integral of f over the interval [pic]is defined as [pic]. In the case where the limit exists, the improper integral is said to converge, and the limit is defined to the value of the integral. In the case where the limit does not exist, the improper integral is said to diverge, and it is not assigned a value.

(b) The improper integral of f over the interval [pic] is defined as [pic]. The integral is said to converge if the limit exists and diverge if it does not.

(c) The improper integral of f over the interval [pic] is defined as [pic] where c is any real number. The improper integral is said to converge if both terms converge and diverge if either term diverges.

[pic]

[pic]

3. Integrals whose integrands have infinite discontinuities:

(a) If f is continuous on the interval [a, b], except for an infinite discontinuity at b, then the improper integral of f over the interval [a, b] is defined as [pic]. In the case where the limit exists, the improper integral is said to converge, and the limit is defined to be the value of the integral. In the case where the limit does not exist, the improper integral is said to diverge, and it is not assigned a value.

[pic]

(b) If f is continuous on the interval [a, b], except for an infinite discontinuity at a, then the improper integral of f over the interval [a, b] is defined as [pic]. The integral is said to converge if the limit exists and diverge if it does not.

[pic]

(c) If f is continuous on the interval [a, b], except for an infinite discontinuity at a point c in (a, b), then the improper integral of f over the interval [a, b] is defined as [pic]. The improper integral is said to converge if both terms converge and diverge if either term diverges.

[pic]

[pic]

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