What is the integral of 4xcot(x^2) ?

The integral of 4xcot(x^2) is 2ln|sin(x^2)|+C

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integral of 4xcot(x^2)

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

2 ln sin (x^{2}) + C

Solution steps

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

Take the constant out:

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

= 4 \cdot \int x cot (x^{2}) d x

Apply u-substitution

: \int \frac{cot ( u )}{2} d u

= 4 \cdot \int \frac{cot ( u )}{2} d u

Take the constant out:

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

= 4 \cdot \frac{1}{2} \cdot \int cot (u) d u

Rewrite using trig identities

= 4 \cdot \frac{1}{2} \cdot \int \frac{cos ( u )}{sin ( u )} d u

Apply u-substitution

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

= 4 \cdot \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 4 \cdot \frac{1}{2} ln ∣ v ∣

Substitute back

= 4 \cdot \frac{1}{2} ln sin (x^{2})

Simplify

4 \cdot \frac{1}{2} ln sin (x^{2}) : 2 ln sin (x^{2})

= 2 ln sin (x^{2})

Add a constant to the solution

= 2 ln sin (x^{2}) + C

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What is the integral of 4xcot(x^2) ?
The integral of 4xcot(x^2) is 2ln|sin(x^2)|+C