What is the integral of 3xcot(x^2) ?

The integral of 3xcot(x^2) is 3/2 ln|sin(x^2)|+C

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integral of 3xcot(x^2)

\int 3 x cot (x^{2}) d x

\frac{3}{2} ln sin (x^{2}) + C

Solution steps

\int 3 x cot (x^{2}) d x

Take the constant out:

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

= 3 \cdot \int x cot (x^{2}) d x

Apply u-substitution

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

= 3 \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

= 3 \cdot \frac{1}{2} \cdot \int cot (u) d u

Rewrite using trig identities

= 3 \cdot \frac{1}{2} \cdot \int \frac{cos ( u )}{sin ( u )} d u

Apply u-substitution

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

= 3 \cdot \frac{1}{2} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= 3 \cdot \frac{1}{2} ln ∣ v ∣

Substitute back

= 3 \cdot \frac{1}{2} ln sin (x^{2})

Simplify

3 \cdot \frac{1}{2} ln sin (x^{2}) : \frac{3}{2} ln sin (x^{2})

= \frac{3}{2} ln sin (x^{2})

Add a constant to the solution

= \frac{3}{2} ln sin (x^{2}) + C

\frac{d}{d θ} (a sec^{2} (\frac{θ}{2}))

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\int (4 x^{3} - 2)^{2} d x

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\frac{d}{d x} (x^{6} - 1)

What is the integral of 3xcot(x^2) ?
The integral of 3xcot(x^2) is 3/2 ln|sin(x^2)|+C