Question

\(\int(5-4x)e^{-x}dx=\)

• A. e^-x(4x-9)+C
• B. e^-x(9-4x)+C
• C. e^-x(4x-5)+C
• D. e^-x(4x-1)+C
• E. e^-x(5-4x)+C

Answer 1

The Correct Option is A

We are given the integral \( \int (5 - 4x)e^{-x} \, dx \) and asked to find the solution. 

We can solve this using the method of integration by parts. Let:

\( u = 5 - 4x \), so that \( du = -4 \, dx \),

and let \( dv = e^{-x} \, dx \), so that \( v = -e^{-x} \).

Using the integration by parts formula \( \int u \, dv = uv - \int v \, du \), we get:

\( \int (5 - 4x) e^{-x} \, dx = -(5 - 4x) e^{-x} - \int (-e^{-x})(-4) \, dx \).

Simplifying this, we get:

\( -(5 - 4x) e^{-x} + 4 \int e^{-x} \, dx \).

The integral of \( e^{-x} \) is \( -e^{-x} \), so we have:

\( -(5 - 4x) e^{-x} - 4e^{-x} + C \),

which simplifies to:

\( e^{-x}(4x - 9) + C \).

The correct answer is \( e^{-x}(4x - 9) + C \).