Answered: "Evaluate :∫ctg3x71-cos2x"

Guadalupe Glass

Answered question

2022-04-03

Evaluate :
$\int \frac{\sqrt[7]{c t g^{3} (x)}}{1 - \cos^{2} x}$

pypberissootcu

Beginner2022-04-04Added 14 answers

You mean

\int \frac{\cot^{\frac{3}{7}} (x)}{1 - \cos^{2} (x)} d x

Consider the substitution

u = \cot (x), d u = \frac{- d x}{\sin^{2} (x)}

. Then

\int \frac{\cot^{\frac{3}{7}} (x)}{1 - \cos^{2} (x)} d x = \int \frac{\cot^{\frac{3}{7}} (x)}{\sin^{2} (x)} d x

= \int - u^{\frac{3}{7}} d u

= \frac{- 7}{10} \cot^{\frac{10}{7}} (x) + C

Also, you can't really do this without (at least implicitly) using the

\csc (x)

function. The cosecant function is defined as

\csc (x) = \frac{1}{\sin (x)}

, so as long as you have the definition of

\sin (x)

, you also have the definition of

\csc (x)

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