Answer:

✏️ INTEGRATION :

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We would like to integrate the following Integral:

\displaystyle \int 5 {x}^{4} \, dx∫5x

4

dx

Well, to get the constant we can consider the following Integration rule:

\displaystyle \int c{x} ^{n} \, dx = c\int {x}^{n} \, dx∫cx

n

dx=c∫x

n

dx

Therefore,

\displaystyle 5\int {x}^{4} \, dx5∫x

4

dx

Recall exponent integration rule:

\displaystyle \int {x} ^{n} \, dx = \frac{ {x}^{n + 1} }{n + 1}∫x

n

dx=

n+1

x

n+1

So let,

n = 4n=4

Thus integrate:

\displaystyle = 5\left( \frac{ {x}^{4+ 1} }{4 + 1} \right)=5(

4+1

x

4+1

)

Simplify addition:

\displaystyle = 5\left( \frac{ {x}^{5} }{5} \right)=5(

5

x

5

)

Reduce fraction:

\displaystyle = {x}^{5}=x

5

Finally we of course have to add the constant of integration:

{\small{\underline{\boxed{\rm{\pink{ {x}^{2} + C }}}}}}

x

2

+C

Hence , our answer is D

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Step-by-step explanation:

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