Mathematics: Evaluate exactly the value of the integral from negative 1 to 0 of the product of the cube of the quantity 2 times x to the 4th power plus 8 times x and 4 times x to the 3rd power plus 4, dx. your work must include the use of substitution and the antiderivative.

Evaluate exactly the value of the integral from

Ответ:

emilymartinez75

13.03.2022

$\bf \displaystyle \int_{-1}^{0}~(2x^4+8x)^3(4x^3+4)\cdot dx\\\\
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u=2x^4+8x\implies \cfrac{du}{dx}=8x^3+8\implies \cfrac{du}{2(4x^3+4)}=dx\\\\
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\displaystyle \int_{-1}^{0}~u^3\underline{(4x^3+4)}\cdot \cfrac{du}{2\underline{(4x^3+4)}}\implies \cfrac{1}{2}\int_{-1}^0~u^3\cdot du\\\\
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$

$\bf \textit{now, let's change the bounds, using }u(x)\\\\
u(-1)=2(-1)^4+8(-1)\implies u(-1)=-6
\\\\\\
u(0)=2(0)^4+8(0)\implies u(0)=0\\\\
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\displaystyle \cfrac{1}{2}\int_{-6}^0~u^3\cdot du\implies \left.\cfrac{1}{2}\cdot \cfrac{u^4}{4} \right]_{-6}^0\implies \left. \cfrac{u^4}{8} \right]_{-6}^0
\\\\\\\
[0]~~-~~[162]\implies -162$

notice, by changing the bounds using u(x), we do not need to substitute back the "u".

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Ответ:

deepspy599otchpd

10.04.2020

Mathematics

The equation ofa line that passes throug (x1,y1) and has s slope of m is
y-y1=m(x-x1)

given
point is (-1,-3) and slope=4

y-(-3)=4(x-(-1)) or
y+3=4(x+1)

B is the answer

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Evaluate exactly the value of the integral from negative 1 to 0 of the product of the cube of the quantity 2 times x to the 4th power plus 8 times x and 4 times x to the 3rd power plus 4, dx. your work must include the use of substitution and the antiderivative.

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