Evaluate the following integrals. intfrac{4^{cot x}}{sin^2x}dx

Evaluate the following integrals. intfrac{4^{cot x}}{sin^2x}dx

Question
Integrals
asked 2020-11-20
Evaluate the following integrals.
\(\int\frac{4^{\cot x}}{\sin^2x}dx\)

Answers (1)

2020-11-21
\(\text{Given integral is }\int\frac{4^{\cot x}}{\sin^2x}\)
\(\text{Rewrite the above integrtal as }\int\frac{4^{\cot x}}{\sin^2x}=\int4^{\cot x}\csc^2(x)dx\)
\(\text{Consider }u=\cot x.\)
\(\text{Then,}\)
\(\frac{du}{dx}=-\csc^2x\)
\(dx=-\frac{du}{\csc^2x}\)
\(\text{Substitute }u=\cot x\ and\ dx=-\frac{du}{\csc^2x}\ in\ \int4^{\cot x}\csc^2(x)dx.\)
\(\int4^{\cot x}\csc^2(x)dx=\int4^u\csc^2(x)(-\frac{du}{\csc^2x})\)
\(=-\int4^udu\)
\(=\frac{-4^u}{\ln(4)}+C\)
\(=-\frac{4^{\cot x}}{\ln(4)}+C\)
\(=-\frac{2^{2\cot x}}{\ln(2)^2}+C\)
\(=-\frac{2^{2\cot x}}{2\ln(2)}+C\)
\(=-\frac{2^{2\cot x-1}}{\ln(2)}+C\)
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