What is 31.75 divided by .57

How many applications of integration by parts are required to evaluate integral (x^7)(e^x) dx?

charle [14.2K]

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Make it more general, and see what happens when you try to reduce the exponent of x in the following integral:

$\mathsf{\mathtt{I}_0=\displaystyle\int\! x^k\cdot e^x\,dx\qquad\qquad (k\ge 1,~~k\in\mathbb{N})}$

Now, integrate it by parts:

$\begin{array}{lcl}
\mathsf{u=x^k}&\quad\Rightarrow\quad&\mathsf{du=k\cdot x^{k-1}\,dx}\\\\
\mathsf{dv=e^x\,dx}&\quad\Leftarrow\quad&\mathsf{v=e^x}
\end{array}$

$\mathsf{\displaystyle\int\! u\,dv=u\cdot v-\int\! v\,du}\\\\\\
\mathsf{\displaystyle\int\! x^k\cdot e^x\,dx=x^k\cdot e^x-\int\! e^x\cdot k\cdot x^{k-1}\,dx}\\\\\\
\mathsf{\displaystyle\int\! x^k\cdot e^x\,dx=x^k\cdot e^x-k\int\! x^{k-1}\cdot e^x\,dx}\\\\\\
\mathsf{\displaystyle\int\! x^k\cdot e^x\,dx=x^k\cdot e^x-k\cdot \mathtt{I}_1}$

where $\mathsf{\mathtt{I}_1=\displaystyle\int\! x^{k-1}\cdot e^x\,dx.}$

So after one iteration, the exponent of x was decreased by one unit.

The question is: after how many iterations will the exponent of x equals zero?

After exactly k iterations, of course.

Therefore, for k = 7, you have to apply integration by parts 7 times, to get rid of that polynomial factor. Then, there will be one last integral left to evaluate:

$\mathsf{\displaystyle\int\! e^x\,dx}$

But this one doesn't need to be evaluated by parts. You can directly write the result:

$\mathsf{\displaystyle\int\! e^x\,dx=e^x+C}$

Shortly, for the integral

$\mathsf{\mathtt{I}_0=\displaystyle\int\! x^7\cdot e^x\,dx}$

you have to apply integration by parts 7 times (not 8 times).

I hope this helps. =)

Tags: <em>indefinite integral integration by parts reduction formula product polynomial exponential differential integral calculus</em>

8 0

3 years ago

Solve for x.<br> 3(2x + 5) = 33

Umnica [9.8K]

Answer:

x = 3

Step-by-step explanation:

3(2x + 5) = 33

3 * 2x + 3 * 5 = 33

6x + 15 = 33

6x = 33 - 15

x = 18 / 6

x = 3

6 0

2 years ago

F(z) = (z - 5)(2x + 7) (73 = 3) has zeros at x = -3.5, x=3/7 , x=5 What is the sign of f on the interval 3/7 < x <5?

Gwar [14]

<h3>Answer: Negative</h3>

======================================================

Explanation:

3/7 = 0.42857 approximately

Pick a number between that value and 5, not including either endpoint. Let's say we pick x = 2

Plug x = 2 into the f(x) function

f(x) = (x - 5)(2x + 7)(7x-3)

f(2) = (2 - 5)(2*2 + 7)(7*2-3)

f(2) = (2 - 5)(4 + 7)(14-3)

f(2) = (-3)(11)(11)

f(2) = -363

The actual result doesn't matter. All we're after is whether the result is positive or negative. We see the result is negative. This means f(x) is negative when 3/7 < x < 5. The f(x) curve is below the x axis on this interval.

3 0

3 years ago

Suppose you invest $1400 at interest rate of 4.3% compounded continuously. How much will you have in the account after 7 years?

STatiana [176]

$\bf ~~~~~~ \textit{Continuously Compounding Interest Earned Amount} \\\\ A=Pe^{rt}\qquad \begin{cases} A=\textit{accumulated amount}\\ P=\textit{original amount deposited}\dotfill & \$1400\\ r=rate\to 4.3\%\to \frac{4.3}{100}\dotfill &0.043\\ t=years\dotfill &7 \end{cases} \\\\\\ A=1400e^{0.043\cdot 7}\implies A=1400e^{0.301}\implies A\approx 1891.69\implies \stackrel{\textit{rounded up}}{A=1892}$

3 0

2 years ago

For an outdoor concert by the city orchestra, concert organizers estimate that 10,000 people will attend if it's not raining. If

Vlada [557]

Answer:

The number of expected people at the concert is 8,500 people

Step-by-step explanation:

In this question, we are asked to determine the expected number of people that will attend a concert if we are given the probabilities that it will rain and it will not rain.

We proceed as follows;

The probability that it will rain is 30% or 0:3

The probability that it will not rain would be 1 -0.3 = 0.7

Now, we proceed to calculate the number of people that will attend by multiplying the probabilities by the expected number of people when it rains and when it does not rain.

Mathematically this is;

Number of expected guests = (probability of not raining * number of expected guests when it does not rain) + (probability of raining * number of expected guests when it rains)

Let’s plug values;

Number of expected guests = (0.3 * 5,000) + (0.7 * 10,000) = 1,500 + 7,000 = 8,500 people

4 0

3 years ago