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MAT137 | Calculus! | Lecture 30 Today: Next: Integration Methods: §8.4 Trig. Substitution §8.5 Rational Functions. Integration Methods: §8.3 Trig. Functions official website http://uoft.me/MAT137 Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Trigonometric Identities Pythagorean Identities sin2 x + cos2 x = 1 tan2 x + 1 = sec2 x 1 + cot2 x = csc2 x Derivatives of Trigonometric Functions d tan x = sec2 x dx d sin x = cos x dx d cos x = − sin x dx Beatriz Navarro-Lameda d sec x = sec x tan x dx L0601 MAT137 24 January 2017 Trigonometric Identities Pythagorean Identities sin2 x + cos2 x = 1 (1) tan2 x + 1 = sec2 x (2) 2 2 1 + cot x = csc x (3) Double-Angle Formulas cos(2x) = cos2 x − sin2 x = 1 − 2 sin2 x (4) sin(2x) = 2 sin x cos x (5) Half-Angle Formulas sin2 x = 1 2 − 21 cos(2x) (6) cos2 x = 1 2 + 12 cos(2x) (7) You can obatain all these identities using a clever combination of (1) and (4). Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Example 1 Evaluate ∫ cos8 x sin x dx. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Example 2 Evaluate ∫ cos3 x sin4 x dx. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Example 3 Evaluate ∫ sin5 x cos2 x dx. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Example 4 Evaluate ∫ sin3 x dx. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Method To compute ∫ sinn x cosm x dx: if if , then try u = sin x; , then try u = cos x. If both m and n are odd, you can try either substitution. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Method To compute ∫ sinn x cosm x dx: if m is odd, then try u = sin x; if , then try u = cos x. If both m and n are odd, you can try either substitution. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Method To compute ∫ sinn x cosm x dx: if m is odd, then try u = sin x; if n is odd , then try u = cos x. If both m and n are odd, you can try either substitution. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Example 5 Evaluate ∫ cos2 x dx. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Example 6 Evaluate ∫ sin4 x dx. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions We can use the same strategy to compute integrals involving powers of tan x and sec x. Homework: Try to evaluate the following integrals. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Example 7 Evaluate ∫ sec4 x tan5 x dx. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Example 8 Evaluate ∫ sec3 x tan3 x dx. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Method To compute ∫ secn x tanm x dx: if m is odd, then try Beatriz Navarro-Lameda ; L0601 MAT137 24 January 2017 Integral of Trig. Functions Method To compute ∫ secn x tanm x dx: if m is odd, then try ; → save a factor of sec x tan x and use tan2 x = sec2 x − 1 to express the remaining factors in terms of sec x. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017 Integral of Trig. Functions Method To compute ∫ secn x tanm x dx: if m is odd, then try ; → save a factor of sec x tan x and use tan2 x = sec2 x − 1 to express the remaining factors in terms of sec x. if n is even, then try Beatriz Navarro-Lameda . L0601 MAT137 24 January 2017 Integral of Trig. Functions Method To compute ∫ secn x tanm x dx: if m is odd, then try ; → save a factor of sec x tan x and use tan2 x = sec2 x − 1 to express the remaining factors in terms of sec x. if n is even, then try . → save a factor of sec2 x and use sec2 x = tan2 x + 1 to express the remaining factors in terms of tan x. Beatriz Navarro-Lameda L0601 MAT137 24 January 2017