Evaluate the integral.
-$$\int x \csc ^ { 2 } 3 x d x$$
A) $- \frac { 1 } { 3 } x \cot 3 x + \frac { 1 } { 9 } \ln | \sin 3 x | + C$
B) $- x \cot 3 x + \ln | \sin 3 x | + C$
C) $\frac { 1 } { 3 } x \cot 3 x - \frac { 1 } { 9 } \ln | \sin 3 x | + C$
D) $- 3 x \cot 3 x + 9 \ln | \sin 3 x | + C$

Question

Quizplus · Accepted Answer

This integral can be solved using integration by parts, where $u = x$ and $dv = \csc^2 3x dx$. The derivative of $u$ is $du = dx$, and the integral of $dv$ is $v = -\frac{1}{3}\cot 3x$. Applying the integration by parts formula, $\int u dv = uv - \int v du$, gives $-\frac{1}{3}x\cot 3x + \frac{1}{9}\ln|\sin 3x| + C$.

Evaluate the Integral ∫xcsc⁡23xdx\int x \csc ^ { 2 } 3 x d x∫xcsc23xdx

Evaluate the Integral $\int x \csc ^ { 2 } 3 x d x$