Evaluate the Integral integral of 1/(e^(2x)) with respect to x

Problem

(∫_^)(1/(e(2*x))*d(x))

Solution

1. Rewrite the integrand using negative exponents to make it easier to integrate.

1/(e(2*x))=e(−2*x)

2. Identify the integral form for an exponential function with a linear exponent, which is (∫_^)(e(a*x)*d(x))=1/a*e(a*x)+C

a=−2

3. Apply the integration rule by dividing by the coefficient of x and keeping the exponential term.

(∫_^)(e(−2*x)*d(x))=(e(−2*x))/(−2)+C

4. Simplify the expression by moving the negative sign and optionally rewriting the exponent as a positive power in the denominator.

−1/(2*e(2*x))+C

Final Answer

(∫_^)(1/(e(2*x))*d(x))=−1/(2*e(2*x))+C

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