5int(1)/(1-sin x)dx=

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5int(1)/(1-sin x)dx=>

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Trig Identity Simplification: We are given the integral to solve: ∫ 5/(1 - sin(x)) dx To solve this integral, we can use a trigonometric identity to simplify the integrand. The identity we will use is: sin^2(x) + cos^2(x) = 1 This identity allows us to express sin(x) in terms of cos(x), which can be useful for simplification.Pythagorean Identity: We can rewrite the denominator 1 - sin(x) using the Pythagorean identity by adding and subtracting cos^2(x) to the denominator: 1 - sin(x) = 1 - sin^2(x) + cos^2(x) - cos^2(x) = (1 - sin^2(x)) + cos^2(x) - cos^2(x) = cos^2(x) + (1 - sin^2(x)) - cos^2(x) = cos^2(x) + cos^2(x) - cos^2(x) = cos^2(x) Now we have: ∫ 5/(1 - sin(x)) dx = ∫ 5/cos^2(x) dxWeierstrass Substitution: To simplify the integral, we can use the Weierstrass substitution, which involves the substitution: t = tan(x/2) Then we have: sin(x) = 2t/(1 + t^2) cos(x) = (1 - t^2)/(1 + t^2) dx = 2/(1 + t^2) dt Now we can rewrite the integral in terms of t.Integrand Simplification: Substituting the expressions for sin(x), cos(x), and dx into the integral, we get: ∫ 5/(1 - sin(x)) dx = ∫ 5/(1 - 2t/(1 + t^2)) * (2/(1 + t^2)) dt Now we need to simplify the integrand.Standard Integral Formula: Simplify the integrand: = ∫ 5 * (1 + t^2)/(1 + t^2 - 2t) * (2/(1 + t^2)) dt = ∫ 5 * 2/(1 + t^2 - 2t) dt = ∫ 10/(1 - 2t + t^2) dt = ∫ 10/(t - 1)^2 dt Now we have a simpler integral to solve.Substitution for t: The integral ∫ 10/(t - 1)^2 dt is a standard integral that can be solved using the formula: ∫ a/(x - b)^2 dx = -a/(x - b) + C where a is a constant, b is the constant term in the denominator, and C is the constant of integration. Applying this formula, we get: ∫ 10/(t - 1)^2 dt = -10/(t - 1) + CFinal Answer: Now we need to substitute back for t to express the answer in terms of x. Recall that t = tan(x/2), so we have: -10/(t - 1) + C = -10/(tan(x/2) - 1) + C This is the antiderivative in terms of x.

$5\int \frac{1}{1-\sin x}\,dx\Rightarrow$

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$5\int \frac{1}{1-\sin x}\,dx\Rightarrow$