7int(1+sin(x))/(1-sin(x))dx

Identify Integral: Identify the integral to be solved. We need to find the integral of the function (1+sin(x))/(1-sin(x)) with respect to x.Simplify Integrand: Simplify the integrand if possible. In this case, we can try to simplify the integrand by using a trigonometric identity or by manipulating the expression. However, the expression does not simplify easily using basic trigonometric identities.Use Substitution: Use substitution to simplify the integral. Let's use the substitution method. We can let u = 1 - sin(x), which means du = -cos(x)dx. We need to express everything in the integral in terms of u, including dx.Find dx: Find dx in terms of du. Since du = -cos(x)dx, we can solve for dx. dx = -du/cos(x).Substitute u and dx: Substitute u and dx into the integral. Substituting u and dx into the integral, we get: ∫(1+sin(x))/(1-sin(x))dx = ∫(1+sin(x))/u * (-du/cos(x))Express sin(x): Express sin(x) in terms of u. We have u = 1 - sin(x), so sin(x) = 1 - u. Now we can substitute this into the integral.Substitute sin(x): Substitute sin(x) with 1 - u in the integral. After substitution, the integral becomes: ∫(1+(1-u))/u * (-du/cos(x))Simplify Integrand: Simplify the integrand. Simplify the expression inside the integral: ∫(2-u)/u * (-du/cos(x))Realize Mistake: Realize there is a mistake in the substitution. We made a mistake in the substitution process. The correct substitution should have been: ∫(1+(1-u))/u * (-du/cos(x)) = ∫(2-u)/u * (-du/cos(x)) However, we cannot proceed because we have not expressed cos(x) in terms of u, which is necessary to complete the substitution. We need to go back and correct this.

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$7 \int \frac{1+\sin (x)}{1-\sin (x)} d x$

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Algebra 1

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7 \int \frac{1+\sin (x)}{1-\sin (x)} d x

Identify Integral: Identify the integral to be solved. $\newline$ We need to find the integral of the function $(1+\sin(x))/(1-\sin(x))$ with respect to $x$ .
Simplify Integrand: Simplify the integrand if possible. $\newline$ In this case, we can try to simplify the integrand by using a trigonometric identity or by manipulating the expression. However, the expression does not simplify easily using basic trigonometric identities.
Use Substitution: Use substitution to simplify the integral. $\newline$ Let's use the substitution method. We can let $u = 1 - \sin(x)$ , which means $du = -\cos(x)dx$ . We need to express everything in the integral in terms of $u$ , including $dx$ .
Find $dx$ : Find $dx$ in terms of $du$ . $\newline$ Since $du = -\cos(x)dx$ , we can solve for $dx$ . $dx = -\frac{du}{\cos(x)}$ .
Substitute $u$ and $dx$ : Substitute $u$ and $dx$ into the integral. $\newline$ Substituting $u$ and $dx$ into the integral, we get: $\newline$ $\int\frac{1+\sin(x)}{1-\sin(x)}dx = \int\frac{1+\sin(x)}{u} \cdot \left(-\frac{du}{\cos(x)}\right)$
Express $\sin(x)$ : Express $\sin(x)$ in terms of $u$ . We have $u = 1 - \sin(x)$ , so $\sin(x) = 1 - u$ . Now we can substitute this into the integral.
Substitute $\sin(x)$ : Substitute $\sin(x)$ with $1 - u$ in the integral. $\newline$ After substitution, the integral becomes: $\newline$ $\int\frac{1+(1-u)}{u} \cdot \left(-\frac{du}{\cos(x)}\right)$
Simplify Integrand: Simplify the integrand. $\newline$ Simplify the expression inside the integral: $\newline$ $\int\frac{2-u}{u} \cdot \frac{-\mathrm{d}u}{\cos(x)}$
Realize Mistake: Realize there is a mistake in the substitution. $\newline$ We made a mistake in the substitution process. The correct substitution should have been: $\newline$ $\int(1+(1-u))/u * (-du/\cos(x)) = \int(2-u)/u * (-du/\cos(x))$ $\newline$ However, we cannot proceed because we have not expressed $\cos(x)$ in terms of $u$ , which is necessary to complete the substitution. We need to go back and correct this.