Resolved: "Evaluate the following integrals. ∫4cot⁡xsin2⁡xdx"

banganX

Answered question

2020-11-20

Evaluate the following integrals.

\int \frac{4^{\cot x}}{\sin^{2} x} d x

lamusesamuset

Skilled2020-11-21Added 93 answers

Given integral is \int \frac{4^{\cot x}}{\sin^{2} x}

Rewrite the above integrtal as \int \frac{4^{\cot x}}{\sin^{2} x} = \int 4^{\cot x} \csc^{2} (x) d x

Consider u = \cot x .

Then,

\frac{d u}{d x} = - \csc^{2} x

d x = - \frac{d u}{\csc^{2} x}

Substitute u = \cot x a n d d x = - \frac{d u}{\csc^{2} x} i n \int 4^{\cot x} \csc^{2} (x) d x .

\int 4^{\cot x} \csc^{2} (x) d x = \int 4^{u} \csc^{2} (x) (- \frac{d u}{\csc^{2} x})

= - \int 4^{u} d u

= \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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