question evaluate the integral below. ∫2x csc²(x) dx be sure to place the argument of any trigonometric or…

question evaluate the integral below. ∫2x csc²(x) dx be sure to place the argument of any trigonometric or logarithmic functions in parentheses in your answer. provide your answer below: ∫2x csc²(x) dx =

question evaluate the integral below. ∫2x csc²(x) dx be sure to place the argument of any trigonometric or logarithmic functions in parentheses in your answer. provide your answer below: ∫2x csc²(x) dx =

Answer

Explanation:

Step1: Apply integration - by - parts formula

The integration - by - parts formula is $\int u;dv=uv-\int v;du$. Let $u = 2x$ and $dv=\csc^{2}(x)dx$. Then $du = 2dx$ and $v=-\cot(x)$.

Step2: Substitute into the formula

$\int 2x\csc^{2}(x)dx=2x(-\cot(x))-\int(-\cot(x))\times2dx$.

Step3: Simplify the first term and solve the second integral

The first term is $- 2x\cot(x)$. The second integral $\int2\cot(x)dx = 2\int\frac{\cos(x)}{\sin(x)}dx$. Let $t=\sin(x)$, then $dt=\cos(x)dx$. So $2\int\frac{\cos(x)}{\sin(x)}dx = 2\ln|\sin(x)|+C$.

Answer:

$-2x\cot(x)+2\ln|\sin(x)|+C$