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

Question

Quizplus · Accepted Answer

To solve the integral $\int x \csc^2 3x \, dx$, we use integration by parts, where $u = x$ and $dv = \csc^2 3x \, dx$. This gives $du = dx$ and $v = -\frac{1}{3}\cot 3x$, since the integral of $\csc^2 x$ is $-\cot x$. Applying the integration by parts formula $\int u \, dv = uv - \int v \, du$, we get $-\frac{1}{3} x \cot 3x - \int -\frac{1}{3} \cot 3x \, dx$. The integral of $\cot x$ is $\ln |\sin x|$, so the final answer is $-\frac{1}{3} x \cot 3x + \frac{1}{9} \ln |\sin 3x| + C$, which matches choice B.

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