Final answer to the problem
$\ln\left(\sqrt{2}\right)-\ln\left(\sqrt{\left(\tan\left(\frac{x}{2}\right)+1\right)^2-2}\right)+\frac{x}{2}+\frac{1}{2}\ln\left(\sec\left(\frac{x}{2}\right)^2\right)+C_2$
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Step-by-step Solution
Learn how to solve problems step by step online. Solve the trigonometric integral int((1-tan(x))^(-1))dx. Applying the property of exponents, \displaystyle a^{-n}=\frac{1}{a^n}, where n is a number. We can solve the integral \int\frac{1}{1-\tan\left(x\right)}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. Substituting in the original integral we get.
Final answer to the problem
$\ln\left(\sqrt{2}\right)-\ln\left(\sqrt{\left(\tan\left(\frac{x}{2}\right)+1\right)^2-2}\right)+\frac{x}{2}+\frac{1}{2}\ln\left(\sec\left(\frac{x}{2}\right)^2\right)+C_2$
