Evaluate the Integral

Problem

(∫_0^1)(e^(−4*x)*d(x))

Solution

1. Identify the integral as a definite integral of an exponential function of the form e^(a*x)

2. Apply the integration rule for exponential functions, which states that (∫_^)(e^(a*x)*d(x))=1/a*e^(a*x)+C

3. Find the antiderivative of e^(−4*x) by setting a=−4

F(x)=−1/4*e^(−4*x)

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit 1 and the lower limit 0

[−1/4*e^(−4*x)]^1_0

5. Substitute the limits into the expression.

−1/4*e^(−4*(1))−(−1/4*e^(−4*(0)))

6. Simplify the expression using the fact that e^0=1

−1/4*e^(−4)+1/4

7. Factor out the common term to reach the final form.

1/4*(1−e^(−4))

Final Answer

(∫_0^1)(e^(−4*x)*d(x))=(1−e^(−4))/4

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