What is the integral of 5csc(x) ?

The integral of 5csc(x) is 5ln|tan(x/2)|+C

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integral of 5csc(x)

\int 5 csc (x) d x

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

Solution steps

\int 5 csc (x) d x

Take the constant out:

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

= 5 \cdot \int csc (x) d x

Apply u-substitution

= 5 \cdot \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= 5 ln ∣ u ∣

Substitute back

u = tan (\frac{x}{2})

= 5 ln tan (\frac{x}{2})

Add a constant to the solution

= 5 ln tan (\frac{x}{2}) + C

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