Home > Calculus > Evaluating the Constant of Integration

What is #f(x) = int e^(3x)-e^(x)dx# if #f(0)=-2 #?

Answer 1

I found: #f(x)=e^(3x)/3-e^x-4/3#

We can solve the integral and write: #f(x)=inte^(3x)dx-inte^xdx=e^(3x)/3-e^x+c# now we need to find #c#; we use the fact that #f(0)=-2#, i.e., we use #x=0# and set it equal to #-2#: #e^(3*0)/3-e^0+c=-2# #1/3-1+c=-2# #c=1-2-1/3=-4/3# giving your final function as: #f(x)=e^(3x)/3-e^x-4/3#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Charles Anctil

To find the function ( f(x) = \int e^{3x} - e^x , dx ) given ( f(0) = -2 ), we first integrate ( e^{3x} - e^x ) with respect to ( x ) to find the antiderivative:

[ F(x) = \frac{e^{3x}}{3} - e^x + C ]

where ( C ) is the constant of integration. Then, using the given initial condition ( f(0) = -2 ), we substitute ( x = 0 ) into ( F(x) ) and equate it to ( -2 ) to find the value of ( C ).

[ F(0) = \frac{e^{3(0)}}{3} - e^0 + C = \frac{1}{3} - 1 + C = -2 ]

Solving for ( C ):

[ \frac{1}{3} - 1 + C = -2 \implies C = -2 + 1 - \frac{1}{3} = -\frac{8}{3} ]

Thus, the function ( f(x) = \int e^{3x} - e^x , dx ) with ( f(0) = -2 ) is given by:

[ f(x) = \frac{e^{3x}}{3} - e^x - \frac{8}{3} ]

By signing up, you agree to our Terms of Service and Privacy Policy

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.