What is the integral of arccsc(nt) ?

The integral of arccsc(nt) is tarccsc(nt)+1/n ln|nt+sqrt(n^2t^2-1)|+C

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integral of arccsc(nt)

\int arccsc (n t) d t

t arccsc (n t) + \frac{1}{n} ln n t + n^{2} t^{2} - 1 + C

Solution steps

\int arccsc (n t) d t

Apply u-substitution

= \int \frac{arccsc ( u )}{n} d u

Take the constant out:

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

= \frac{1}{n} \cdot \int arccsc (u) d u

Apply Integration By Parts

= \frac{1}{n} (u arccsc (u) - \int - \frac{u}{u ^{2} u ^{2} - 1} d u)

\int - \frac{u}{u ^{2} u ^{2} - 1} d u = - ln u + u^{2} - 1

= \frac{1}{n} (u arccsc (u) - (- ln u + u^{2} - 1))

Substitute back

u = n t

= \frac{1}{n} (n t arccsc (n t) - (- ln n t + (n t)^{2} - 1))

Simplify

\frac{1}{n} (n t arccsc (n t) - (- ln n t + (n t)^{2} - 1)) : t arccsc (n t) + \frac{1}{n} ln n t + n^{2} t^{2} - 1

= t arccsc (n t) + \frac{1}{n} ln n t + n^{2} t^{2} - 1

Add a constant to the solution

= t arccsc (n t) + \frac{1}{n} ln n t + n^{2} t^{2} - 1 + C

What is the integral of arccsc(nt) ?
The integral of arccsc(nt) is tarccsc(nt)+1/n ln|nt+sqrt(n^2t^2-1)|+C