Question

7–84. Evaluate the following integrals.45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx

Accepted Answer

Step 1: Recognize that the integral involves the function $$ \frac{1}{(1 + e^x)^2} . $$This suggests that substitution might simplify the problem. Look for a substitution that simplifies the denominator $$ 1 + e^x . $$Step 2: Let $$ u = 1 + e^x . $$Then, differentiate $$ u $$ with respect to $$ x $$: $$ \frac{du}{dx} = e^x $$, or equivalently $$ du = e^x dx . $$Rewrite $$ e^x dx  in $$terms of $$ u . $$Step 3: Substitute $$ u $$ into the integral. When $$ x = 0 $$, $$ u = 1 + e^0 = 2 . $$When $$ x = \ln 2 $$, $$ u = 1 + e^{\ln 2} = 1 + 2 = 3 . $$The integral becomes $$ \int_{2}^{3} \frac{1}{u^2} \cdot \frac{du}{e^x} . $$Since $$ e^x = u - 1 $$, replace $$ e^x $$ with $$ u - 1 . $$Step 4: Simplify the integral to $$ \int_{2}^{3} \frac{1}{u^2(u - 1)} du . $$This integral can be solved using partial fraction decomposition. Express $$ \frac{1}{u^2(u - 1)}  as$$ $$ \frac{A}{u} + \frac{B}{u^2} + \frac{C}{u - 1} $$, and solve for $$ A $$, $$ B $$, and $$ C . $$Step 5: After finding the coefficients $$ A $$, $$ B $$, and $$ C $$, rewrite the integral as a sum of simpler integrals: $$ \int \frac{A}{u} du + \int \frac{B}{u^2} du + \int \frac{C}{u - 1} du . $$Evaluate each term separately and apply the limits of integration $$ u = 2  to$$ $$ u = 3 .$$

7–84. Evaluate the following integrals.
45. ∫ from 0 to ln 2 [1 / (1 + eˣ)²] dx

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