Recall from Math 141 Integration Basics Fundamental Theorem of
Transcription
Recall from Math 141 Integration Basics Fundamental Theorem of
Recall from Math 141 Integration Basics Fundamental Theorem of Calculus (FTC) Let f : [a, b] → R be a continuous function. Let F : [a, b] → R be a function. • If F is an antiderivative of f on [a, b] (i.e. F 0 (x) = f (x) for each x ∈ [a, b]), then Z b Z b f (x) dx ≡ F 0 (x) dx = F (b) − F (a) . a a Rx • If F (x) = a f (t) dt for each x ∈ [a, b], then F is an antiderivative of f on [a, b], i.e. Z x 0 F (x) ≡ Dx f (t) dt = f (x) . a Basic Differentiation Rules If the functions y = f (x) and y = g(x) are differentiable at x and a and b are constants, then: (1) Dx [af (x) + bg(x)] = af 0 (x) + bg 0 (x) (2) Dx h[f (x)i · g(x)] = f 0 (x) · g(x) + f (x) · g 0 (x) 0 (x) − f (x) · g 0 (x) (3) Dx fg(x) = f (x) · g(x)[g(x)] provided g(x) 6= 0 . 2 If f is differentiable at x and g is differentiable at f (x), then: (4) Dx [g(f (x))] = g 0 (f (x)) f 0 (x) . On this handout, a (∗) means that you do not have to memorize the formula but should be able to use the formula if you are given it. Hyperbolic Trig Functions1 ex + e−x 2 (∗) sinh x tanh x = cosh x (∗) (∗) cosh x = sinh x = (∗) coth x = derivatives cosh x sinh x ex − e−x 2 (∗) cosh2 x − sinh2 x = 1 (∗) sech x = FTC =⇒ du dx du (∗) Dx sinh u = cosh u dx 1 du (∗) Dx sinh−1 u = √ 2 1 + u dx 1 du (∗) Dx cosh−1 u = √ 2 u>1 u − 1 dx Dx cosh u = sinh u 1Hyperbolic (∗) csch x = 1 sinh x integrals Z (∗) 1 cosh x (∗) sinh u du = cosh u + C Z (∗) cosh u du = sinh u + C Z du (∗) −1 u √ = sinh +C a a2 + u2 a>0 Z du (∗) −1 u √ = cosh +C a u2 − a2 u>a>0 Trig Functions are covered in §7.3, pages 439–447. 16.08.17 (yr.mn.dy) Page 1 of 4 Integration Recall from Math 141 Integration Basics Basic Integral Formulas FTC =⇒ derivatives Dx un = nun−1 integrals Z du dx du Dx e = e dx u6=0 1 du Dx ln |u| = u dx du Dx au = au ln a 0 < a 6= 1 dx 1 du u6=0 1 Dx loga |u| = 0 < a 6= 1 u ln a dx du Dx sin u = cos u dx du Dx tan u = sec2 u dx du Dx sec u = sec u tan u dx du Dx cos u = − sin u dx du Dx cot u = − csc2 u dx du Dx csc u = − csc u cot u dx 1 du Dx sin−1 u = √ = −Dx cos−1 u 2 1 − u dx 1 du Dx tan−1 u = = −Dx cot−1 u 2 1 + u dx 1 du √ Dx sec−1 u = = −Dx csc−1 u 2 |u| u − 1 dx more integrals u u Z un du n6=−1 = un+1 + C n+1 eu du = eu + C Z du u6=0 = ln |u| + C u Z au au du = + C ln a Z cos u du = sin u + C Z sec2 u du = tan u + C Z sec u tan u du = sec u + C Z sin u du = − cos u + C Z csc2 u du = − cot u + C Z csc u cot u du = − csc u + C Z du u a>0 √ + C = sin−1 2 2 a a −u Z du u a>0 1 tan−1 + C = 2 2 a +u a a Z du |u| a>0 1 √ = sec−1 +C 2 2 a a u u −a Z tan u du = − ln |cos u| + C = ln |sec u| + C cot u du = ln |sin u| + C = − ln |csc u| + C sec u du = ln |sec u + tan u| + C = − ln |sec u − tan u| + C csc u du = − ln |csc u + cot u| + C = ln |csc u − cot u| + C Z Z Z 16.08.17 (yr.mn.dy) Page 2 of 4 Integration Recall from Math 141 Integration Basics Generalized Exponential y = bx and Logarithmic y = logb x Functions with base b where b > 0 but b 6= 1. Also 0 < a 6= 1. ln ≡ loge f (x) = bx ≡ ex ln b : (−∞, ∞) → (0, ∞) g(x) = logb x ≡ the inverse of the fn. f (x) = b y = logb x ⇐⇒ (loga b)(logb c) = loga c x : (0, ∞) → (−∞, ∞) x = by ln x loga x = ln a =⇒ x, y > 0 & r ∈ R x, y ∈ R & r ∈ R blogb x = x logb (bx ) = x logb 1 = 0 b0 = 1 logb (xy) = logb x + logb y x = logb x − logb y logb y logb (xr ) = r(logb x) bx by = bx+y bx = bx−y by (bx )r = bxr x x x (ab) = a b and a x b ax = x b Basic Trig hypotenuse cos θ = adj hyp sin θ = opp hyp tan θ = opp adj cot x = cos x sin x sec x = 1 cos x csc x = 1 sin x opposite θ adjacent tan x = sin x cos x Basic Inverse Trig Functions y = sin θ ⇔ θ = sin−1 y where −1≤y ≤1 and y = cos θ ⇔ θ = cos−1 y where −1≤y ≤1 and y = tan θ ⇔ θ = tan−1 y where y∈R and y = cot θ ⇔ θ = cot−1 y where y∈R and y = sec θ ⇔ θ = sec−1 y where |y| ≥ 1 and y = csc θ ⇔ θ = csc−1 y where |y| ≥ 1 and 16.08.17 (yr.mn.dy) Page 3 of 4 −π π ≤θ≤ 2 2 0≤θ≤π −π π <θ< 2 2 0<θ<π π 0 ≤ θ ≤ π , θ 6= 2 −π π ≤ θ ≤ , θ 6= 0 2 2 Integration To come in Math 142 Integration Basics Our Math 142 Course Homepage (CH) is http://people.math.sc.edu/girardi/w142.html Integration by Parts Z Z u dv = uv − v du Trig Identities useful for Integration Double-Angle Formulas: cos(2x) = cos2 x − sin2 x sin(2x) = 2 sin x cos x Half-Angle Formulas: cos2 x = 1 + cos(2x) 2 sin2 x = 1 − cos(2x) 2 Addition/Subtraction Formulas: (∗) (∗) cos(s + t) = cos s cos t − sin s sin t sin(s + t) = sin s cos t + cos s sin t (∗) (∗) cos(s − t) = cos s cos t + sin s sin t sin(s − t) = sin s cos t − cos s sin t Trig Substitution if integrand involves a2 − u 2 a2 + u 2 u 2 − a2 then make the substitution u u = a sin θ ! θ = sin−1 a −1 u u = a tan θ ! θ = tan a −1 u u = a sec θ ! θ = sec a restriction on θ −π π ≤θ≤ 2 2 −π π <θ< 2 2 π 0 ≤ θ ≤ π , θ 6= 2 Space for your personal notes. 16.08.17 (yr.mn.dy) Page 4 of 4 Integration
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