Evaluate the Integral

Problem

(∫_^)((e(2*x))/(1+e(2*x))*d(x))

Solution

1. Identify a suitable substitution to simplify the integrand, noting that the numerator is a multiple of the derivative of the denominator.

2. Substitute u=1+e(2*x) into the integral.

3. Differentiate u with respect to x to find d(u)=2*e(2*x)*d(x)

4. Rearrange the differential to solve for the terms in the numerator: 1/2*d(u)=e(2*x)*d(x)

5. Rewrite the integral in terms of u

(∫_^)(1/u⋅1/2*d(u))

6. Factor out the constant:

1/2*(∫_^)(1/u*d(u))

7. Integrate using the natural logarithm rule:

1/2*ln(u)+C

8. Back-substitute u=1+e(2*x) to return to the original variable x Since 1+e(2*x) is always positive, the absolute value bars can be changed to parentheses.

Final Answer

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

Want more problems? Check here!