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Solve the Definite Integral: int_(-6)^(14)10x^(4)+12x^(3)-16x^(2)-9x+15 dx Section 5.4

Solve the Definite Integral:\newline∫−61410x4+12x3−16x2−9x+15dx \int_{-6}^{14} 10 x^{4}+12 x^{3}-16 x^{2}-9 x+15 d x \newlineSection 55.44

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Q. Solve the Definite Integral:\newline∫−61410x4+12x3−16x2−9x+15dx \int_{-6}^{14} 10 x^{4}+12 x^{3}-16 x^{2}-9 x+15 d x \newlineSection 55.44
  1. Calculate F(14)F(14): Now we'll evaluate the antiderivative from −6-6 to 1414.
    F(b)−F(a)=[2(14)5+3(14)4−(16/3)(14)3−(9/2)(14)2+15(14)]−[2(−6)5+3(−6)4−(16/3)(−6)3−(9/2)(−6)2+15(−6)]F(b) - F(a) = [2(14)^5 + 3(14)^4 - (16/3)(14)^3 - (9/2)(14)^2 + 15(14)] - [2(-6)^5 + 3(-6)^4 - (16/3)(-6)^3 - (9/2)(-6)^2 + 15(-6)]
    Let's calculate F(14)F(14) first.
    F(14)=2(14)5+3(14)4−(16/3)(14)3−(9/2)(14)2+15(14)F(14) = 2(14)^5 + 3(14)^4 - (16/3)(14)^3 - (9/2)(14)^2 + 15(14)
  2. Calculate F(−6)F(-6): Now let's calculate F(−6)F(-6).F(−6)=2(−6)5+3(−6)4−(163)(−6)3−(92)(−6)2+15(−6)F(-6) = 2(-6)^5 + 3(-6)^4 - \left(\frac{16}{3}\right)(-6)^3 - \left(\frac{9}{2}\right)(-6)^2 + 15(-6)
  3. Subtract F(−6)F(-6) from F(14)F(14): Subtract F(−6)F(-6) from F(14)F(14) to get the value of the definite integral.\newline∫−614(10x4+12x3−16x2−9x+15) dx=F(14)−F(−6)\int_{-6}^{14} (10x^4 + 12x^3 - 16x^2 - 9x + 15)\,dx = F(14) - F(-6)
  4. Correct antiderivative calculation: Oops, I just realized I made a mistake in the antiderivative calculation. The antiderivative of 10x410x^4 should be (105)x5(\frac{10}{5})x^5, which is 2x52x^5, not 10x510x^5. I need to correct this before proceeding.

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