What is the integral of csc(3x) ?

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

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integral of csc(3x)

\int csc (3 x) d x

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

Solution steps

\int csc (3 x) d x

Apply u-substitution

: \int csc (u) \frac{1}{3} d u

= \int csc (u) \frac{1}{3} d u

Take the constant out:

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

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

Apply u-substitution

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

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

Use the common integral:

\int \frac{1}{v} d v = ln (∣ v ∣)

= \frac{1}{3} ln ∣ v ∣

Substitute back

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

Add a constant to the solution

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

y^{^{'}} = y - y^{2}, y (0) = 9

\frac{2}{( s + 2 ) ( s + 3 ) ^{2}}

\int 5 x e^{x} d x

\int_{\frac{π}{6}}^{\frac{π}{4}} tan (x) d x

\frac{1}{s ^{4} ( s ^{2} - 4 )}

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