$\text { Find } \int \cot ^ { 3 } x \csc ^ { 19 } x d x \text {. }$
A) $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 19 } \csc ^ { 19 } x - \frac { 1 } { 20 } \csc ^ { 20 } x + C$
B) $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 21 } \csc ^ { 21 } x - \frac { 1 } { 19 } \csc ^ { 19 } x + C$
C) $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 21 } \csc ^ { 21 } x - \frac { 1 } { 20 } \csc ^ { 20 x } x + C$
D) $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 20 } \csc ^ { 20 } x - \frac { 1 } { 19 } \csc ^ { 19 } x + C$
E) $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 19 } \csc ^ { 19 } x - \frac { 1 } { 21 } \csc ^ { 21 } x + C$

Question

$\text { Find } \int \cot ^ { 3 } x \csc ^ { 19 } x d x \text {. }$ 
A) $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 19 } \csc ^ { 19 } x - \frac { 1 } { 20 } \csc ^ { 20 } x + C$
B) $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 21 } \csc ^ { 21 } x - \frac { 1 } { 19 } \csc ^ { 19 } x + C$
C) $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 21 } \csc ^ { 21 } x - \frac { 1 } { 20 } \csc ^ { 20 x } x + C$
D) $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 20 } \csc ^ { 20 } x - \frac { 1 } { 19 } \csc ^ { 19 } x + C$
E) $\int \cot ^ { 3 } x \csc ^ { 19 } x d x = \frac { 1 } { 19 } \csc ^ { 19 } x - \frac { 1 } { 21 } \csc ^ { 21 } x + C$

Quizplus · Accepted Answer

The integral of $\cot^3 x \csc^{19} x dx$ involves a reduction formula that typically results in terms involving decreasing powers of $\csc x$. The correct answer involves a combination of $\csc^{19} x$ and $\csc^{21} x$ with their coefficients arranged to reflect the integration process, which involves integrating by parts or using trigonometric identities to simplify the expression. The given options suggest a mix-up in the signs and coefficients, but the correct approach should yield a result that involves both $\csc^{19} x$ and $\csc^{21} x$ with appropriate coefficients, as seen in option E.

Find ∫cot⁡3xcsc⁡19xdx. \text { Find } \int \cot ^ { 3 } x \csc ^ { 19 } x d x \text {. } Find ∫cot3xcsc19xdx.

$\text { Find } \int \cot ^ { 3 } x \csc ^ { 19 } x d x \text {. }$