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What is #f(x) = int (x-3)^2-3x+4 dx# if #f(2) = 4 #?

Answer 1

#7/3#

#int(x-3)^2-3x+4 dx# #= (x-3)^3/3-(3x^2)/2+4x+C# #= (2-3)^3/3-(3(2)^2)/2+4(2)+C = 4# #= -1/3-6+8+C = 4# #= 5/3+C = 4# #C = 7/3#

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Answer 2

Charles Arocha

#f(x)=1/3x^3-9/2x^2+13x-20/3#

#"expand "(x-3)^2" and simplify"#

#f(x)=int(x^2-6x+9-3x+4)dx#

#color(white)(f(x))=int(x^2-9x+13)dx#

#"integrate each term using the "color(blue)"power rule"#

#•color(white)(x)int(ax^n)dx=a/(n+1)x^(n+1);n!=-1#

#rArrf(x)=1/3x^3-9/2x^2+13x+c#

#"where c is the constant of variation"#

#"to find c use the condition "(f(2)=4#

#rArr4=8/3-18+26+crArrc=-20/3#

#rArrf(x)=1/3x^3-9/2x^2+13x-20/3#

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Answer 3

Evelyn Agard

To find ( f(x) = \int (x-3)^2 - 3x + 4 , dx ) if ( f(2) = 4 ), we integrate the given function with respect to ( x ) and then use the given information to solve for the constant of integration. Given that ( f(2) = 4 ), we substitute ( x = 2 ) into the function and solve for the constant. After finding the constant, we integrate the function with the determined constant to find the definite integral.

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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.