integral table math contests

8/12/2019 Integral Table Math Contests

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Integral Table for Math Contests

Section I. Direct Integration

I. II.

III. IV. V.

VI. VII.

VIII. IX. X.

XI. XII.

XIII. XIV. XV.

XVI. XVII.

XVIII.


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With the previous integrals you can derive the next ones.

XIX.

XX.

XXI.

XXII.

Section II. Integration by substitution

XXIII. Use substitution: XXIV. Use substitution:

XXV. Use substitution:

XXVI. Use substitution:

XXVII. Use substitution:


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Section III. Integration by parts

Tabular integrals are of the form: Cyclic integrals are of the form: Other integrals:

For finding reduction formulas of trigonometric functions like , factorizeappropriately ( , make equal to term with the greatestexponent, apply the formula of integration by parts and solve the cyclic integral.

For integrals of the form: use better product-to-sum trigonometricidentities.

Direct cyclic integrals:

XXVIII. + CXXIX. XXX.

XXXI. XXXII.

Section IV. Integrals containing a quadratic trinomial

For all the next integrals it is necessary to reduce the quadratic trinomial to its perfectsquare form: . Based upon this:

XXXIII. If l < 0 then

If l > 0 then


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XXXIV.

XXXV. Substitute with , simplify and the integral is reduced tothe XXXIV formula.

XXXVI. Reduce the trinomial to its perfect square form and use formulas

XXIV or XXVI.

Section V. Rational Functions

Integrals of the form , where the order of is n and the order of is m ,

If first a polynomial division is necessary, then the integral can be solved.

If partial fractions can be used or the Ostrogradsky method in case the roots of

have multiplicity greater than one. If the polynomial has complex roots of multiplicity k , then terms of the form

can be found in the denominator. If the

multiplicity of this terms is one, the fraction is integrated directly; if it is greater than one,

represent the quadratic trinomial in the form and make

the substitution .

Some useful integrals:

XXXVII.

XXXVIII.

If a > 0 then

If a < 0 then


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XXXIX. 2 arctan 2 1

XL.

XLI. (Ostrogradsky method)

is the greatest common divisor of the polynomial and its derivative .

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and are polynomials with undetermined coefficients. and are computed by differentiating the initial equation.

Section VI. Irrational functions

XLII. Substitute with

n is the least common multiple of the numbers q1 , q2, etc.

XLIII.

is a polynomial of degree n-1 with undetermined coefficients.

is real number.

The coefficients and are found by differentiating the above equality.

XLIV. Substitute with , and use formula XLIII.XLV.

If p is an integer number. Expand the binomial or make appropriate substitution.

If is an integer number. Substitution: , s is the denominator of p .

If is an integer number. Substitution: , s is the denominator of p .


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Section VII. Trigonometric functions

XLVI. If m and/or n are odd, use the Pythagorean identity: .

If m and n are even, use the trigonometric power reduction formulas:

XLVII.

If m is odd, express the integral in terms of and make the next substitutions

. If n is even, express the integral in terms of and make the next substitutions

.

XLVIII. Use Pythagorean formulas:

XLIX. Solve with the next formulas:

L. Use substitution If , use substitution

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