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4 years ago

Expand and combine like terms. (2x^4+3x^3)(2x^4-3x^3)

Mathematics

1 answer:

Juli2301 [7.4K]4 years ago

7 0

Step-by-step explanation:

Use the form (a + b)(a - b) = a^2 - b^2:

(2x^4 + 3x^3)(2x^4 - 3x^3) = 4x^8 - 9x^6.

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Evaluate using integration by parts

PolarNik [594]

Rather than carrying out IBP several times, let's establish a more general result. Let

$I(n)=\displaystyle\int x^ne^x\,\mathrm dx$

One round of IBP, setting

$u=x^n\implies\mathrm du=nx^{n-1}\,\mathrm dx$

$\mathrm dv=e^x\,\mathrm dx\implies v=e^x$

gives

$\displaystyle I(n)=x^ne^x-n\int x^{n-1}e^x\,\mathrm dx$

$I(n)=x^ne^x-nI(n-1)$

This is called a power-reduction formula. We could try solving for $I(n)$ explicitly, but no need. $n=5$ is small enough to just expand $I(5)$ as much as we need to.

$I(5)=x^5e^x-5I(4)$

$I(5)=x^5e^x-5(x^4e^x-4I(3))=(x-5)x^4e^x+20I(3)$

$I(5)=(x-5)x^4e^x+20(x^3e^x-3I(2))=(x^2-5x+20)x^3e^x-60I(2)$

$I(5)=(x^2-5x+20)x^3e^x-60(x^2e^x-2I(1))=(x^3-5x^2+20x-60)x^2e^x+120I(1)$

$I(5)=(x^3-5x^2+20x-60)x^2e^x+120(xe^x-I(0))$

Finally,

$I(0)=\displaystyle\int e^x\,\mathrm dx=e^x+C$

so we end up with

$I(5)=(x^4-5x^3+20x^2-60x+120)xe^x-120e^x+C$

$I(5)=(x^5-5x^4+20x^3-60x^2+120x-120)e^x+C$

and the antiderivative is

$\displaystyle\int2x^5e^x\,\mathrm dx=(2x^5-10x^4+40x^3-120x^2+240x-240)e^x+C$

8 0

3 years ago

What is the volume of the pyramid?

Roman55 [17]

Answer:

A. 144 ft^3

Step-by-step explanation:

5 0

3 years ago

Each month, Molly budgets $1450 for fixed expenses, $600 for living expenses, and $340 for annual expenses. Her annual net incom

den301095 [7]

Answer:

surplus of 150$

Step-by-step explanation:

1853.68 + 178. times .249 = ......................owed at the end of the month

1450 +600 + 340 = $2390. * 12= $28 680. annually ..

30480 - 28680 = $1800. surplus at the end of the year divided by 12 = $150.00 monthly surplus

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3 years ago

Read 2 more answers

The linear function f(x) = 0 5x + 80 represents the average test score in your math class, where x is the number of the test tak

SashulF [63]

Answer:

more information

Step-by-step explanation:

6 0

3 years ago

I need help, please I have until 9 p.m. to finish it number 3 homework

sergeinik [125]

Answer: 1/3

<u>Step-by-step explanation:</u>

When adding or subtracting fractions, they must have the same denominator.

$\dfrac{5}{9}-\boxed{\dfrac{?}{?}}=\dfrac{12}{54}\\\\\\\dfrac{5}{9}\bigg(\dfrac{6}{6}\bigg)-\boxed{\dfrac{?}{?}}=\dfrac{12}{54}\\\\\\\dfrac{30}{54}-\boxed{\dfrac{?}{?}}=\dfrac{12}{54}\\\\\\\dfrac{30}{54}-\boxed{\dfrac{18}{54}}=\dfrac{12}{54}\\\\\\\text{Simplify}\quad \dfrac{18}{54}\div \dfrac{18}{18}\quad =\large\boxed{\dfrac{1}{3}}$

6 0

3 years ago