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(Solved): Evaluate the integral. \int 4cot^(2)(x)csc^(4)(x)dx Consider the shown work. \int 4cot^(2)(x)csc^(4) ...
Evaluate the integral.
\int 4cot^(2)(x)csc^(4)(x)dxConsider the shown work.
\int 4cot^(2)(x)csc^(4)(x)dx=\int 4cot^(2)(x)(1-cot^(2)(x))csc^(2)(x)dx=\int 4(cot^(2)(x)-cot^(4)(x))csc^(2)(x)dxLet
u=cot(x)and
du=-csc^(2)(x)dx.
\int 4(cot^(2)(x)-cot^(4)(x))csc^(2)(x)dx=-4\int (u^(2)-u^(4))du
=-(4)/(3)u^(3)+(4)/(5)u^(5)+C
=-(4)/(3)cot^(3)(x)+(4)/(5)cot^(5)(x)+CIdentify the error in the work shown. The substitution
csc^(2)(x)=1-cot^(2)(x)is not a correct trigonometric identity. The power rule for integrals was not applied correctly. No errors exist in the work shown. When applying the substitution method, an inappropriate expression was used for
u. The resulting integral after applying the substitution method is incorrect. Evaluate the integral.
\int 4cot^(2)(x)csc^(4)(x)dx(Express numbers in exact form. Use symbolic notation and fractions as needed. Use
Cto represent the arbitrary constant.)
\int 4cot^(2)(x)csc^(4)(x)dx=
◻Question Source: Rogawski 4e Calculus Early Transcendental
