Question

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

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).