Evaluate the Integral integral of 5e^(-2x) with respect to x

Problem

(∫_^)(5*e(−2*x)*d(x))

Solution

1. Factor out the constant 5 from the integral to simplify the expression.

(∫_^)(5*e(−2*x)*d(x))=5*(∫_^)(e(−2*x)*d(x))

2. Apply the rule for the integral of an exponential function e(a*x) which is (∫_^)(e(a*x)*d(x))=(e(a*x))/a+C

(∫_^)(e(−2*x)*d(x))=(e(−2*x))/(−2)

3. Multiply the result by the constant 5 that was factored out earlier.

5⋅(e(−2*x))/(−2)=−5/2*e(−2*x)

4. Add the constant of integration C to complete the indefinite integral.

−5/2*e(−2*x)+C

Final Answer

(∫_^)(5*e(−2*x)*d(x))=−5/2*e(−2*x)+C

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