Integration Procedures, Study notes of Differential and Integral Calculus

Download Integration Procedures and more Study notes Differential and Integral Calculus in PDF only on Docsity!

Trigonometric Substitutions

𝑎 2 − 𝑢 2 , 𝑙𝑒𝑡 𝑢 = 𝑎𝑠𝑖𝑛𝜃 𝑎 2

• 𝑢 2 , 𝑙𝑒𝑡 𝑢 = 𝑎𝑡𝑎𝑛𝜃 𝑢 2 − 𝑎 2 , 𝑙𝑒𝑡 𝑢 = 𝑎𝑠𝑒𝑐𝜃

Each of the above substitutions reduces the corresponding radical to the following rational trigonometric expressions. 𝑎 2 − 𝑎 2 𝑠𝑖𝑛 2 𝜃 = 𝑎 1 − 𝑠𝑖𝑛 2 𝜃 = 𝑎 cos 𝜃 𝑎 2

• 𝑎 2 𝑡𝑎𝑛 2 𝜃 = 𝑎 1 + 𝑡𝑎𝑛 2 𝜃 = 𝑎 sec 𝜃 𝑎 2 𝑠𝑒𝑐 2 𝜃 − 𝑎 2 = 𝑎 𝑠𝑒𝑐 2 𝜃 − 1 = 𝑎 tan 𝜃

න 𝑑𝑥 𝑥 2

• 2𝑥 𝒍𝒏 𝒙 + 𝟏 + (𝒙 + 𝟏) 𝟐 −𝟏 + 𝑪

න 𝑑𝑥 𝑥 2 4 − 𝑥 2 − 𝟒 − 𝒙 𝟐 𝟒𝒙

• 𝑪

න 𝑑𝑧 𝑧 2 𝑧 2 − 4 𝒛 𝟐 − 𝟒 𝟒𝒛

• 𝑪

Integration of Rational Fractions

Rational fractions – fractions in which the numerators and denominators are polynomials in the variable of integration. Improper fraction – the degree of polynomial in the numerator is not less than that in the denominator, and such a fraction may always be reduced by division to a mixed fraction consisting of a polynomial and a proper fraction.

We assume that all fractions considered are irreducible , that is, the numerator and denominator have no common factor. 𝑥 2

• 1 𝑥+ 1 3 is a proper, irreducible, rational fraction.

𝟏 𝒂𝒙 + 𝒃

𝟐 (𝒂𝒙 + 𝒃) 𝟐

𝒏 𝒂𝒙 + 𝒃 𝒏

where 𝑨 𝟏

𝟐

𝒏 are constants and 𝑨 𝒏

III. If a quadratic factor 𝒂𝒙 𝟐

• 𝒃𝒙 + 𝒄 occurs once as factor of the denominator, there corresponds to this factor one partial fraction 𝑨𝒙+𝑩 𝒂𝒙 𝟐 +𝒃𝒙+𝒄 , where A and B are constants and 𝑨𝒙 + 𝑩 ≠ 𝟎.

IV. If a quadratic factor 𝒂𝒙 𝟐

• 𝒃𝒙 + 𝒄 occurs n times as factor of the denominator, there corresponds to this factor n partial fractions 𝑨 𝟏

𝟏 𝒂𝒙 𝟐

• 𝒃𝒙 + 𝒄

𝟐

𝟐 (𝒂𝒙 𝟐

• 𝒃𝒙 + 𝒄) 𝟐

𝒏

𝒏 (𝒂𝒙 𝟐

• 𝒃𝒙 + 𝒄) 𝒏