Student scored in the 83rd

Mathematics

1 answer:

melamori03 [73]2 years ago

5 0

Answer:b

Step-by-step explanation:

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

charle [14.2K]

If you're using the app, try seeing this answer through your browser: brainly.com/question/2860229

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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: indefinite integral integration by parts reduction formula product polynomial exponential differential integral calculus

8 0

3 years ago

The house was bought for $ 500 and sold for $ 700 what percentage profit did they gain?

jonny [76]

Answer: 40%

Step-by-step explanation:

700 - 500 = 200

200 / 500 = 40%

6 0

2 years ago

Whens my birthday sept 18 oct 16 jan 3 feb 14

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