int(cos xdx)/(sqrt(sin^(2)x+4sin x-1))

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Identify Integral: Identify the integral to be solved. I = ∫(cos x)/(sqrt(sin^2(x) + 4sin x - 1)) dxSimplify Expression: Simplify the expression inside the square root if possible. The expression under the square root is a quadratic in terms of sin(x), which can be factored or completed to a square if it represents a perfect square trinomial.Factor Quadratic: Attempt to factor the quadratic expression sin^2(x) + 4sin x - 1. The quadratic does not factor easily into real factors, so we consider completing the square to transform it into a form that might be more easily integrated. sin^2(x) + 4sin x - 1 = (sin(x) + 2)^2 - 5Rewrite Integral: Rewrite the integral with the completed square. I = ∫(cos x)/(sqrt((sin(x) + 2)^2 - 5)) dxLook for Substitution: Look for a substitution that will simplify the integral. Let u = sin(x) + 2, then du = cos(x) dx. This substitution will simplify the integral.Perform Substitution: Perform the substitution. I = ∫(1)/(sqrt(u^2 - 5)) duRecognize Standard Form: Recognize the integral as a standard form. The integral ∫(1)/(sqrt(u^2 - a^2)) du is a standard form that corresponds to the inverse hyperbolic function arcsinh(u/a) or the logarithmic form ln|u + sqrt(u^2 - a^2)|.Integrate Using Form: Integrate using the standard form. I = ln|u + sqrt(u^2 - 5)| + C, where C is the constant of integration.Substitute Back: Substitute back to the original variable. I = ln|sin(x) + 2 + sqrt((sin(x) + 2)^2 - 5)| + CSimplify Final Answer: Simplify the final answer if possible. I = ln|sin(x) + 2 + sqrt(sin^2(x) + 4sin(x) - 1)| + C

$\int \frac{\cos x d x}{\sqrt{\sin ^{2} x+4 \sin x-1}}$

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∫cos⁡xdxsin⁡2x+4sin⁡x−1 \int \frac{\cos x d x}{\sqrt{\sin ^{2} x+4 \sin x-1}} ∫sin2x+4sinx−1​cosxdx​

$\int \frac{\cos x d x}{\sqrt{\sin ^{2} x+4 \sin x-1}}$