determine the following.\\n\\(\\int 7e^{-0.4x} \\, dx\\)\\n\\(\\int 7e^{-0.4x} \\, dx = \\square\\) (type an…

Explanation:

Step1: Identify the integral type

This is an integral of an exponential function. Recall the formula for integrating ( e^{ax} ), which is ( \int e^{ax} dx = \frac{1}{a}e^{ax} + C ) (where ( C ) is the constant of integration, and we can ignore it for the antiderivative here as we need an exact answer). Also, we have a constant factor 7, so we can use the constant multiple rule for integration: ( \int kf(x) dx = k\int f(x) dx ), where ( k = 7 ) and ( f(x)=e^{-0.4x} ).

Step2: Apply the constant multiple rule

First, factor out the constant 7: ( \int 7e^{-0.4x} dx = 7\int e^{-0.4x} dx ).

Step3: Integrate the exponential function

Let ( u = -0.4x ), then ( du = -0.4 dx ), so ( dx = \frac{du}{-0.4} ). But we can also use the formula for ( \int e^{ax} dx ) directly with ( a = -0.4 ). The integral of ( e^{ax} ) is ( \frac{1}{a}e^{ax} ), so for ( a = -0.4 ), we have ( \int e^{-0.4x} dx = \frac{1}{-0.4}e^{-0.4x} + C ).

Step4: Multiply by the constant factor

Now, multiply by 7: ( 7\times\frac{1}{-0.4}e^{-0.4x} + C ). Simplify ( \frac{7}{-0.4} ). Let's convert 0.4 to a fraction, ( 0.4=\frac{2}{5} ), so ( \frac{7}{-\frac{2}{5}} = 7\times(-\frac{5}{2})=-\frac{35}{2} ). So the integral becomes ( -\frac{35}{2}e^{-0.4x} + C ). But since we just need the antiderivative (exact answer, we can write it as ( -\frac{35}{2}e^{-0.4x} ) (or we can also write -17.5( e^{-0.4x} ), but as a fraction, ( -\frac{35}{2}e^{-0.4x} ) is exact). Alternatively, we can write 0.4 as ( \frac{2}{5} ), so ( -0.4 = -\frac{2}{5} ), then ( \frac{1}{-0.4}=\frac{1}{-\frac{2}{5}} = -\frac{5}{2} ), so ( 7\times(-\frac{5}{2})e^{-0.4x}=-\frac{35}{2}e^{-0.4x} ).

Answer:

( -\frac{35}{2}e^{-0.4x} + C ) (but since the problem says "type an exact answer" and in the box, we can write the antiderivative without the constant as ( -\frac{35}{2}e^{-0.4x} ) or simplified further. Alternatively, 0.4 is ( \frac{2}{5} ), so -0.4 is ( -\frac{2}{5} ), and ( \frac{7}{-\frac{2}{5}} = -\frac{35}{2} ), so the exact answer is ( -\frac{35}{2}e^{-0.4x} + C ), but if we ignore the constant (since it's an indefinite integral and we need an exact answer, the antiderivative is ( -\frac{35}{2}e^{-0.4x} + C ), but in the box, we can write ( -\frac{35}{2}e^{-0.4x} ) (or ( -17.5e^{-0.4x} ), but fraction is exact).