QuestionJuly 17, 2025

Use any basic integration formula or formulas to find the indefinite integral.(Use C for the constant of integration.) int (4)/(1+e^-4x)dx square

Use any basic integration formula or formulas to find the indefinite integral.(Use C for the constant of integration.) int (4)/(1+e^-4x)dx square
Use any basic integration formula or formulas to find the indefinite integral.(Use C for the constant of integration.) int (4)/(1+e^-4x)dx square

Solution
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Answer

-\ln|1 + e^{-4x}| + C Explanation 1. Simplify the integrand Rewrite \frac{4}{1+e^{-4x}} as 4 \cdot \frac{1}{1+e^{-4x}}. 2. Use substitution Let u = 1 + e^{-4x}, then du = -4e^{-4x}dx. Solve for dx: dx = \frac{du}{-4e^{-4x}}. 3. Substitute and simplify Substitute u and dx into the integral: \int \frac{4}{u} \cdot \frac{du}{-4e^{-4x}}. Simplify to \int -\frac{1}{u} du. 4. Integrate The integral of -\frac{1}{u} is -\ln|u| + C. 5. Back-substitute Replace u with 1 + e^{-4x}: -\ln|1 + e^{-4x}| + C.

Explanation

1. Simplify the integrand<br /> Rewrite $\frac{4}{1+e^{-4x}}$ as $4 \cdot \frac{1}{1+e^{-4x}}$.<br />2. Use substitution<br /> Let $u = 1 + e^{-4x}$, then $du = -4e^{-4x}dx$. Solve for $dx$: $dx = \frac{du}{-4e^{-4x}}$.<br />3. Substitute and simplify<br /> Substitute $u$ and $dx$ into the integral: $\int \frac{4}{u} \cdot \frac{du}{-4e^{-4x}}$. Simplify to $\int -\frac{1}{u} du$.<br />4. Integrate<br /> The integral of $-\frac{1}{u}$ is $-\ln|u| + C$.<br />5. Back-substitute<br /> Replace $u$ with $1 + e^{-4x}$: $-\ln|1 + e^{-4x}| + C$.
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