$$(\cot x)'=-\csc^2x,$$
$$\cot^2x=\csc^2x-1$$
ve
$$(\csc^2x)'=-\cot x\csc x$$
eşitliklerini biliyoruz.
$$\int (4\cot^3x+\cot^2x+\cot x-2)e^xdx$$
$$=\int e^x[\cot x(4\cot^2 x+1)+\cot^2x-2]dx$$
$$=\int e^x[\cot x(4\csc^2 x-3)+\csc^2x-3]dx$$
$$=\int e^x[4\cot x\csc^2 x-3\cot x+\csc^2x-3]dx$$
$$=\int e^x[(4\cot x\csc^2 x-2\csc^ 2x)-(3\cot x+3\csc^2x)-3]dx$$
$$=\int d(-2e^x\csc^2x)-3\int d(e^x\cot x)-3e^x$$
$$=-2e^x\csc^2x-3e^x\cot x-3e^x$$
$$=-e^x(2\csc^2x+3\cot x+3)$$
$$=-e^x(2\cot^2x+3\cot x+5)+c$$