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Question: \[\int_{}^{}{\mathbf{e}^{\mathbf{x}}\mathbf{(1}\mathbf{-}\mathbf{\cot}\mathbf{x}\mathbf{+}\mathbf{\c...

∫ex(1−cot⁡x+cot⁡2x)dx\int_{}^{}{\mathbf{e}^{\mathbf{x}}\mathbf{(1}\mathbf{-}\mathbf{\cot}\mathbf{x}\mathbf{+}\mathbf{\cot}^{\mathbf{2}}\mathbf{x}}\mathbf{)dx}

A

excot⁡x+ce^{x}\cot x + c

B

excos⁡⥂ecx+ce^{x}\cos ⥂ ecx + c

C

−excot⁡x+c- e^{x}\cot x + c

D

excos⁡⥂ecx+ce^{x}\cos ⥂ ecx + c

Answer

−excot⁡x+c- e^{x}\cot x + c

Explanation

Solution

∫ex(cos⁡⥂ec2x−cot⁡x)dx=∫ex[−cot⁡x+cosec2x]=−excot⁡x+c\int_{}^{}{e^{x}(\cos ⥂ ec^{2}x - \cot x)dx} = \int_{}^{}{e^{x}\lbrack - \cot x + \text{cose}\text{c}^{2}x\rbrack} = - e^{x}\cot x + c