How do you evaluate the indefinite integral #int (x^2-x+5)dx#?

Answer 1

#int(x^2-x+5)dx=x^3/3-x^2/2+5x+C#

Remember that an indefinite integral is an antiderivative, and since the derivative of sums is the sum of derivatives then

#intf(x)+g(x)+h(x)dx=intf(x)dx+intg(x)dx+inth(x)dx#

it is the same for integrals

In this case

#f(x)=x^2#
#g(x)=-x#
#h(x)=5#
Since #intx^ndx=x^(n+1)/(n+1)#
#intf(x)dx=intx^2dx=x^3/3#
and also since #int-xdx=-intxdx#
#intg(x)dx=int(-x)dx=-intxdx=-x^2/2#
and since for a constant #c#, #intcdx=cx# then
#inth(x)dx=int5dx=5x#

Then put them together

#int(x^2-x+5)dx=x^3/3-x^2/2+5x+C#
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Answer 2

I tried this:

We can break it into three parts and write: #intx^2dx-intxdx+int5dx=#
we now use the general integration formula as: #color(red)(intx^ndx=x^(n+1)/(n+1)+c)#
we can write our integral as: #intx^2dx-intxdx+5intx^0dx=# and we get: #=x^3/3-x^2/2+5x+c#
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Answer 3

To evaluate the indefinite integral ∫(x^2 - x + 5) dx, you integrate each term separately with respect to x.

∫x^2 dx = (1/3)x^3 + C
∫-x dx = -(1/2)x^2 + C
∫5 dx = 5x + C

Putting it all together:

∫(x^2 - x + 5) dx = (1/3)x^3 - (1/2)x^2 + 5x + C

Where C is the constant of integration.

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Answer 4

To evaluate the indefinite integral ( \int (x^2 - x + 5) , dx ), integrate each term of the polynomial separately. The integral of ( x^2 ) is ( \frac{1}{3}x^3 ), the integral of ( -x ) is ( -\frac{1}{2}x^2 ), and the integral of ( 5 ) is ( 5x ). Combine these results to find the indefinite integral of the given expression. Therefore, ( \int (x^2 - x + 5) , dx = \frac{1}{3}x^3 - \frac{1}{2}x^2 + 5x + C ), where ( C ) is the constant of integration.

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Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.

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