What is the integral of-csc(x) ?

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

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integral of-csc(x)

\int - csc (x) d x

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

Solution steps

\int - csc (x) d x

Take the constant out:

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

= - \int csc (x) d x

Apply u-substitution

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

= - \int \frac{1}{u} d u

Use the common integral:

\int \frac{1}{u} d u = ln (∣ u ∣)

= - ln ∣ u ∣

Substitute back

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

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

Add a constant to the solution

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

\frac{\partial}{\partial x} (ln (y) + 2 x y^{3})

y^{^{''}} + 9 y = 3 x e^{2 x}

t \to 0 lim (\frac{e ^{5 t} - 1}{t})

\int f^{- 1} (x) d x

4 cos (\frac{2 π}{3})