Carl Friedrich Gauss proved the Fundamental Theorem of Algebra which gave us the following result: Every...

 

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Carl Friedrich Gauss proved the Fundamental Theorem of Algebra which gave us the following result: Every polynomial can be factored (over the real numbers) into a product of linear and irreducible (unfactorable) quadratic polynomials. Jamal argues that this result means we can develop a strategy for integrating a subset of rational functions P(x) of this we only need to know how to deal with partial fractions of Q(x) the form: . If the degree of P(z) 2 degree of Q(x), we can try to use long division and then some method of partial fractions on the remainder. • If Q(z) contains a repeated linear factor, I(n), our partial fractions will include terms of the form • If Q() is the product of distinct linear factors, terms of the form 2r dr 3 x² + 4x + 13 P(x) A A₂ Q(x) l(r) (l(x))² • If Q(2) contains an irreducible quadratic factor, g(r), then our partial fractions will include terms of the form r³-r dr + Identify which Jamal's method(s), if any, you would use to integrate the following integrals. Be sure to provide reasoning for your choices. dr P(x) A₁ + Q(x) 4₁(2) 1₂(x) = A₂ (l(x))" (r), our partial fractions will include P(z) Ar+B Q(x) (d) dr (e) +12r³+15² + 25x +11 T³ + 12x² +11z 2x²-x-1 (F) (²+1)(x-2) dz dz Carl Friedrich Gauss proved the Fundamental Theorem of Algebra which gave us the following result: Every polynomial can be factored (over the real numbers) into a product of linear and irreducible (unfactorable) quadratic polynomials. Jamal argues that this result means we can develop a strategy for integrating a subset of rational functions P(x) of this we only need to know how to deal with partial fractions of Q(x) the form: . If the degree of P(z) 2 degree of Q(x), we can try to use long division and then some method of partial fractions on the remainder. • If Q(z) contains a repeated linear factor, I(n), our partial fractions will include terms of the form • If Q() is the product of distinct linear factors, terms of the form 2r dr 3 x² + 4x + 13 P(x) A A₂ Q(x) l(r) (l(x))² • If Q(2) contains an irreducible quadratic factor, g(r), then our partial fractions will include terms of the form r³-r dr + Identify which Jamal's method(s), if any, you would use to integrate the following integrals. Be sure to provide reasoning for your choices. dr P(x) A₁ + Q(x) 4₁(2) 1₂(x) = A₂ (l(x))" (r), our partial fractions will include P(z) Ar+B Q(x) (d) dr (e) +12r³+15² + 25x +11 T³ + 12x² +11z 2x²-x-1 (F) (²+1)(x-2) dz dz Carl Friedrich Gauss proved the Fundamental Theorem of Algebra which gave us the following result: Every polynomial can be factored (over the real numbers) into a product of linear and irreducible (unfactorable) quadratic polynomials. Jamal argues that this result means we can develop a strategy for integrating a subset of rational functions P(x) of this we only need to know how to deal with partial fractions of Q(x) the form: . If the degree of P(z) 2 degree of Q(x), we can try to use long division and then some method of partial fractions on the remainder. • If Q(z) contains a repeated linear factor, I(n), our partial fractions will include terms of the form • If Q() is the product of distinct linear factors, terms of the form 2r dr 3 x² + 4x + 13 P(x) A A₂ Q(x) l(r) (l(x))² • If Q(2) contains an irreducible quadratic factor, g(r), then our partial fractions will include terms of the form r³-r dr + Identify which Jamal's method(s), if any, you would use to integrate the following integrals. Be sure to provide reasoning for your choices. dr P(x) A₁ + Q(x) 4₁(2) 1₂(x) = A₂ (l(x))" (r), our partial fractions will include P(z) Ar+B Q(x) (d) dr (e) +12r³+15² + 25x +11 T³ + 12x² +11z 2x²-x-1 (F) (²+1)(x-2) dz dz