Evaluate the Integral

Problem

(∫_a^b)(5*e(4*t)*d(t))

Solution

1. Identify the constant and the form of the integrand, which is an exponential function of the form e(k*t)

2. Apply the constant multiple rule by moving the factor of 5 outside the integral.

(∫_a^b)(5*e(4*t)*d(t))=5*(∫_a^b)(e(4*t)*d(t))

3. Find the antiderivative of e(4*t) using the rule (∫_^)(e(k*t)*d(t))=1/k*e(k*t)

(∫_^)(e(4*t)*d(t))=1/4*e(4*t)

4. Apply the Fundamental Theorem of Calculus by evaluating the antiderivative at the upper limit b and subtracting the evaluation at the lower limit a

5*[1/4*e(4*t)]ba

5. Substitute the limits into the expression.

5/4*(e(4*b)−e(4*a))

Final Answer

(∫_a^b)(5*e(4*t)*d(t))=5/4*(e(4*b)−e(4*a))

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