Evaluate: `inte^(2x)\ ((1-sin2x)/(1-cos2x))\ dx`

Explanation

Let `I =int e^{2 x}(frac{1-sin 2 x}{1-cos 2 x}) d x `
` I =int e^{2 x}(frac{1-2 sin x cos x}{2 sin ^{2} x}) cdot d x `
`I =int e^{2 x}(frac{1}{2 sin ^{2} x}-frac{2 sin x cos x}{2 sin ^{2} x}) d x`
` I =int e^{2 x}(frac{1}{2} {cosec} x-cot x) d x `
` I =-int e^{2 x}(cot x-frac{1}{2} {cosec}^{2} x) d x `
`p u t 2 x=t Rightarrow x=t / 2 `
`therefore I =-int e^{t}(cot t / 2-frac{1}{2} {cosec} t / 2) frac{d t}{2}`
` I =-frac{1}{2} int e^{t}(cot t / 2-frac{1}{2} {cosec}^{2} t / 2) cdot d t`
` e^{t}(f(t)+f^{prime}(t) cdot d t=e^{t} f(t)+c. `
`f(t)=cot t / 2 f ' t)=-frac{{cosec} t / 2}{2} `
` therefore I .=-frac{1}{2}(e^{t} cot t / 2) cdots+frac{1}{2}+e^{2 x} cot x)+c`
` I =-frac{1}{2}[e^(2x) cot x]+c`