What is the integral of (tan(4cot(x)))/(sin^2(x)) ?

The integral of (tan(4cot(x)))/(sin^2(x)) is 1/4 ln|cos(4cot(x))|+C

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integral of (tan(4cot(x)))/(sin^2(x))

integral

\frac{tan ( 4 cot ( x ) )}{sin ^{2} ( x )}

\frac{1}{4} ln ∣ cos (4 cot (x)) ∣ + C

Solution steps

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

Apply u-substitution

: \int - tan (4 u) d u

= \int - tan (4 u) d u

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - \int tan (4 u) d u

Rewrite using trig identities

= - \int \frac{sin ( 4 u )}{cos ( 4 u )} d u

Apply u-substitution

: \int - \frac{1}{4 v} d v

= - \int - \frac{1}{4 v} d v

Take the constant out:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= - (- \frac{1}{4} \cdot \int \frac{1}{v} d v)

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= - (- \frac{1}{4} ln ∣ v ∣)

Substitute back

= - (- \frac{1}{4} ln ∣ cos (4 cot (x)) ∣)

Simplify

= \frac{1}{4} ln ∣ cos (4 cot (x)) ∣

Add a constant to the solution

= \frac{1}{4} ln ∣ cos (4 cot (x)) ∣ + C

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What is the integral of (tan(4cot(x)))/(sin^2(x)) ?
The integral of (tan(4cot(x)))/(sin^2(x)) is 1/4 ln|cos(4cot(x))|+C