How do you integrate #3x^2-5x+9# from 0 to 7?

Answer 1
You can integrate separately each function and then, once you found the anti-derivative #F(x)#, substitute #0# and #7#: #int_0^7f(x)dx=F(7)-F(0)# In your case: #int_0^7(3x^2-5x+9)dx=# #=int_0^7(3x^2)dx-int_0^7(5x)dx+int_0^7(9)dx=#
Remember that #intkx^ndx=kx^(n+1)/(n+1)+c# where #k# is a constant and #intkdx=kx+c#;

you have:

#=3x^3/3-5x^2/2+9x]_0^7=# you can now substitute the extremes of integration:
#=(7^3-5*7^2/2+9*7)-(0-0+0)=# #=343-122.5+63=283.5#
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Answer 2

To integrate (3x^2 - 5x + 9) from (0) to (7), you can use the definite integral formula:

[ \int_{0}^{7} (3x^2 - 5x + 9) , dx ]

First, find the antiderivative of the function:

[ \int (3x^2 - 5x + 9) , dx = x^3 - \frac{5}{2}x^2 + 9x + C ]

Then, evaluate this antiderivative at the upper and lower limits of integration:

[ \left[ x^3 - \frac{5}{2}x^2 + 9x \right]_0^7 ]

[ = \left(7^3 - \frac{5}{2} \times 7^2 + 9 \times 7 \right) - \left(0^3 - \frac{5}{2} \times 0^2 + 9 \times 0 \right) ]

[ = \left(343 - \frac{5}{2} \times 49 + 63 \right) - (0 - 0 + 0) ]

[ = \left(343 - \frac{245}{2} + 63 \right) - 0 ]

[ = \left(343 - 122.5 + 63 \right) ]

[ = 283.5 ]

So, the definite integral of (3x^2 - 5x + 9) from (0) to (7) is (283.5).

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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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