Final answer to the problem
$\arctan\left(\frac{\sqrt{8\tan\left(\frac{x}{2}\right)^2}}{2\sqrt{2}}\right)-\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)+\frac{128\tan\left(\frac{x}{2}\right)^{3}}{\left(8\tan\left(\frac{x}{2}\right)^2+8\right)^{2}}+C_0$
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Step-by-step Solution
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1
Find the integral
$\int\frac{\sin\left(x\right)^2-\cos\left(x\right)^2}{1-\cot\left(x\right)^2}dx$
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$\int\frac{\sin\left(x\right)^2-\cos\left(x\right)^2}{1-\cot\left(x\right)^2}dx$
Learn how to solve problems step by step online. Integrate the function (sin(x)^2-cos(x)^2)/(1-cot(x)^2). Find the integral. Reduce \frac{\sin\left(x\right)^2-\cos\left(x\right)^2}{1-\cot\left(x\right)^2} by applying trigonometric identities. We can solve the integral \int\frac{\sin\left(x\right)^2-\cos\left(x\right)^2}{1-\cot\left(x\right)^2}dx by applying the method Weierstrass substitution (also known as tangent half-angle substitution) which converts an integral of trigonometric functions into a rational function of t by setting the substitution. Hence.
Final answer to the problem
$\arctan\left(\frac{\sqrt{8\tan\left(\frac{x}{2}\right)^2}}{2\sqrt{2}}\right)-\sin\left(\frac{x}{2}\right)\cos\left(\frac{x}{2}\right)+\frac{128\tan\left(\frac{x}{2}\right)^{3}}{\left(8\tan\left(\frac{x}{2}\right)^2+8\right)^{2}}+C_0$