int(e^(-x)+sin x)/(cos x+e^(-x))dx

Simplify and Factorize: Simplify the integrand if possible. Notice that the numerator and denominator have a common term e^(-x). We can factor this term out to simplify the expression. (e^(-x) + sin(x)) / (cos(x) + e^(-x)) = e^(-x) * (1 + sin(x) * e^(x)) / (e^(-x) * (cos(x) * e^(x) + 1)) After factoring out e^(-x), we get: (1 + sin(x) * e^(x)) / (cos(x) * e^(x) + 1)Recognize Simplified Form: Recognize that the integrand simplifies to a simpler form. After simplification, we see that the e^(-x) terms cancel out, leaving us with: (1 + sin(x) * e^(x)) / (cos(x) * e^(x) + 1) = (1 + sin(x) * e^(x)) / (e^(x) * cos(x) + 1) This simplifies to: 1 / (cos(x) + e^(-x))Integrate Simplified Function: Attempt to integrate the simplified function. The integral of 1 / (cos(x) + e^(-x)) with respect to x is not straightforward and does not correspond to a standard integral form. We need to consider alternative methods such as substitution or partial fractions, but in this case, these methods do not seem to apply. This suggests that the integral may not have a simple closed-form expression.Identify Simplification Mistake: Realize a mistake in the simplification process. Upon reviewing the previous steps, we realize that the simplification in Step 2 was incorrect. The terms e^(x) and e^(-x) do not cancel out in the way described. This is a math error.

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$\int\frac{e^{-x}+\sin x}{\cos x+e^{-x}}\,dx$

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Math Problems

Calculus

Evaluate definite integrals using the chain rule

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\int\frac{e^{-x}+\sin x}{\cos x+e^{-x}}\,dx

Simplify and Factorize: Simplify the integrand if possible. $\newline$ Notice that the numerator and denominator have a common term $e^{-x}$ . We can factor this term out to simplify the expression. $\newline$ $\frac{e^{-x} + \sin(x)}{\cos(x) + e^{-x}} = e^{-x} \cdot \frac{1 + \sin(x) \cdot e^{x}}{e^{-x} \cdot (\cos(x) \cdot e^{x} + 1)}$ $\newline$ After factoring out $e^{-x}$ , we get: $\newline$ $\frac{1 + \sin(x) \cdot e^{x}}{\cos(x) \cdot e^{x} + 1}$
Recognize Simplified Form: Recognize that the integrand simplifies to a simpler form. $\newline$ After simplification, we see that the $e^{-x}$ terms cancel out, leaving us with: $\newline$ $(1 + \sin(x) \cdot e^{x}) / (\cos(x) \cdot e^{x} + 1) = (1 + \sin(x) \cdot e^{x}) / (e^{x} \cdot \cos(x) + 1)$ $\newline$ This simplifies to: $\newline$ $1 / (\cos(x) + e^{-x})$
Integrate Simplified Function: Attempt to integrate the simplified function. $\newline$ The integral of $\frac{1}{\cos(x) + e^{-x}}$ with respect to $x$ is not straightforward and does not correspond to a standard integral form. We need to consider alternative methods such as substitution or partial fractions, but in this case, these methods do not seem to apply. This suggests that the integral may not have a simple closed-form expression.
Identify Simplification Mistake: Realize a mistake in the simplification process. $\newline$ Upon reviewing the previous steps, we realize that the simplification in Step $2$ was incorrect. The terms $e^{x}$ and $e^{-x}$ do not cancel out in the way described. This is a math error.