What is #f(x) = int xe^x-xsqrt(x^2+2)dx# if #f(0)=-2 #?

Answer 1

#f(x)#=#int xe^x*dx#-#int x*sqrt(x^2+2)*dx#

=#x*e^x#-#int e^x*dx#-#1/3*(x^2+2)^(3/2)+C#

=#(x-1)e^x#-#1/3*(x^2+2)^(3/2)+C#

After imposing #f(0)=-2# condition,

#(0-1)*e^0-1/3*2^(3/2)+C=-2#

#C=(2sqrt(2))/3-1#

Hence #f(x)#=#(x-1)e^x#-#1/3*(x^2+2)^(3/2)+(2sqrt(2))/3-1#

  1. I took integral right side.
2) I imposed #f(0)=-2# condition for finding #C#.
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Answer 2

To find the function ( f(x) ), we first differentiate it with respect to ( x ) using the Fundamental Theorem of Calculus. This will yield ( f'(x) = x e^x - x \sqrt{x^2 + 2} ). Then, we integrate ( f'(x) ) to obtain ( f(x) ). Since ( f(0) = -2 ), we substitute ( x = 0 ) into ( f(x) ) and solve for the constant of integration. The integral simplifies to ( \frac{2}{3} (x^3 e^x - (x^3 + 2) \sqrt{x^2 + 2}) ). Substituting ( x = 0 ) gives ( f(0) = \frac{4}{3} ), so the constant of integration is ( -\frac{10}{3} ). Therefore, ( f(x) = \frac{2}{3} (x^3 e^x - (x^3 + 2) \sqrt{x^2 + 2}) - \frac{10}{3} ).

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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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