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    Trig

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    xmikelawsonx
    The document contains information about: 1) Power substitutions that can be used when trigonometric functions are raised to even or odd powers. 2) Square root substitutions that can be used when taking integrals involving square roots. 3) The partial fractions decomposition method for splitting a rational function into partial fractions based on the linear or quadratic factors of the denominator. 4) A list of common integrals and their solutions.

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    Trig

    Uploaded by

    xmikelawsonx
    14 views3 pages
    The document contains information about: 1) Power substitutions that can be used when trigonometric functions are raised to even or odd powers. 2) Square root substitutions that can be used when taking integrals involving square roots. 3) The partial fractions decomposition method for splitting a rational function into partial fractions based on the linear or quadratic factors of the denominator. 4) A list of common integrals and their solutions.

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    The document contains information about: 1) Power substitutions that can be used when trigonometric functions are raised to even or odd powers. 2) Square root substitutions that can be used when taking integrals involving square roots. 3) The partial fractions decomposition method for splitting a rational function into partial fractions based on the linear or quadratic factors of the denominator. 4) A list of common integrals and their solutions.

    Copyright:

    Attribution Non-Commercial (BY-NC)
    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    14 views3 pages

    Trig

    Uploaded by

    xmikelawsonx
    The document contains information about: 1) Power substitutions that can be used when trigonometric functions are raised to even or odd powers. 2) Square root substitutions that can be used when taking integrals involving square roots. 3) The partial fractions decomposition method for splitting a rational function into partial fractions based on the linear or quadratic factors of the denominator. 4) A list of common integrals and their solutions.

    Copyright:

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    Download as PDF, TXT or read online from Scribd
    Download as pdf or txt
    of 3

    Power Substitutions

    sinm x cosn x m or n odd m or n even u = cos x sin2 = 1 (1 cos 2) 2 cos2 = 1 (1 + cos 2) 2 sin mx cos nx
    1 sin A sin B = 2 (cos(A B) cos(A + B))

    cos A cos B = 1 (cos(A B) + cos(A + B)) 2 sin A cos B = 1 (sin(A + B) + sin(A B) 2 tanm x secn x m odd u = tan x n even u = sec x
    1 sin2 = 2 (1 cos 2) 1 cos2 = 2 (1 + cos 2)

    Square Root Substitutions


    Expression in Integral a2 x2 a2 + x2 x2 a2 Substitution x = a sin x = a tan x = a sec Identity 1 sin2 = cos2 1 + tan2 = sec2 sec2 1 = tan2

    Partial Fractions
    P (x) Q(x) Where P (x) and Q(x) are polynomials. If degree of P (x) degree of Q(x), then divide rst.

    Case I: Q(x) is a product of unique linear factors


    Method: P (x) A1 A2 Ak = + + ... + Q(x) a1 x + b1 a2 x + b2 ak x + bk (1)

    Case II: Q(x) factors as a product of (possibly) repeated linear factors.


    Method: For each factor (ai x + bi )ri P (x) A1 A2 Ar i = + + ... + Q(x) ai x + bi (ai x + bi )2 (ai x + bi )ri

    (2)

    Case III: Q(x) factors as a product of unique irreducible quadratic factors.


    ie. (b2 4ai ci ) < 0, i : Z i Method: P (x) A1 x + B1 A2 x + B 2 Ak x + B k = + + ... + Q(x) a1 x2 + b1 x + c1 a2 x2 + b2 x + c ak x2 + bk x + ck (3)

    Case IV: Q(x) is a product of repeated irreducible quadratic factors.


    ie. (b2 4ai ci ) < 0, i : Z i Method: For each factor (ai x + bi )ri P (x) A1 x + B 1 Ari x + Bri Ak x + B k = + + ... + Q(x) ai x2 + bi x + ci (ai x2 + bi x + ci )2 (ai x2 + bi x + ci )ri

    (4)

    Common Integrals
    xn+1 n+1 csc x dx = ln | csc x cot x|

    xn dx =

    1 dx = ln |x| x

    tan x dx = ln | sec x|

    cot x dx = ln | sin x| ex dx = ex

    ax dx =

    ax ln a

    sinh x dx = cosh x

    cosh x dx = sinh x sin x dx = cos x

    cos x dx = sin x

    1 x dx dx = tan1 x2 a2 a a

    sec2 x dx = tan x

    x2

    x dx dx = sin1 2 a a

    csc2 x dx = cot x

    sec x tan x dx = sec x

    csc x cot x dx = csc x

    sec x dx = ln | sec x + tan x|

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