What is the integral of wcsc(4w^2-1) ?

The integral of wcsc(4w^2-1) is 1/8 ln|tan((4w^2-1)/2)|+C

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integral of wcsc(4w^2-1)

\int w csc (4 w^{2} - 1) d w

\frac{1}{8} ln tan (\frac{4 w ^{2} - 1}{2}) + C

Solution steps

\int w csc (4 w^{2} - 1) d w

Apply u-substitution

= \int \frac{csc ( u )}{8} d u

Take the constant out:

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

= \frac{1}{8} \cdot \int csc (u) d u

Apply u-substitution

= \frac{1}{8} \cdot \int \frac{1}{v} d v

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{8} ln ∣ v ∣

Substitute back

= \frac{1}{8} ln tan (\frac{4 w ^{2} - 1}{2})

Add a constant to the solution

= \frac{1}{8} ln tan (\frac{4 w ^{2} - 1}{2}) + C

What is the integral of wcsc(4w^2-1) ?
The integral of wcsc(4w^2-1) is 1/8 ln|tan((4w^2-1)/2)|+C