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

The integral of xcsc(3x^2) is 1/6 ln|tan((3x^2)/2)|+C

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integral of xcsc(3x^2)

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

\frac{1}{6} ln tan (\frac{3 x ^{2}}{2}) + C

Solution steps

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

Apply u-substitution

: \int \frac{csc ( u )}{6} d u

= \int \frac{csc ( u )}{6} d u

Take the constant out:

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

= \frac{1}{6} \cdot \int csc (u) d u

Apply u-substitution

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

= \frac{1}{6} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{6} ln ∣ v ∣

Substitute back

= \frac{1}{6} ln tan (\frac{3 x ^{2}}{2})

Add a constant to the solution

= \frac{1}{6} ln tan (\frac{3 x ^{2}}{2}) + C

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What is the integral of xcsc(3x^2) ?
The integral of xcsc(3x^2) is 1/6 ln|tan((3x^2)/2)|+C