Answer to Question #218959 in Calculus for heron

Question

evaluate

1. $\int$ cscⁿxdx

Answer 1

We have

I_n = $\int cosec^n x \ dx$

I_n = $\int$ cosec^n-2 x cosec² x dx

Now integrating the above expression by by using the formula of $\int$ uv dx.

Considering cosec^n-2 x = u and cosec² x = v. We have,

I_n = cosec^n-2 x $\int$ cosec² x dx - $\int$ [ $\dfrac{d}{dx}$ (cosec^n-2 x) $\int$ cosec² x dx ] dx

I_n = - cosec^n-2 x cot x - $\int$ [ (n-2) cosec^n-3 x cosec x cot² x] dx

I_n = - cosec^n-2 x cot x - $\int$ [ (n-2) cosec^n-2 x (cosec² x - 1) ] dx

I_n = - cosec^n-2 x cot x - (n - 2) $\int$ [ cosecⁿ x - cosec^n-2 x ] dx ...............equation(1)

Noe from the above equation it can be seen that

I_n = $\int$ cosecⁿ x dx and I_n-2 = $\int$ cosec^n-2 x dx .................equation(2)

So on substituting the values from equation(2) to equation(1), we have

I_n = - cosec^n-2 x cot x - (n - 2) [ I_n - I_n-2 ]

I_n (n - 1) = - cosec^n-2 x cot x + (n - 2) I_n-2

I_n = $\dfrac{-1}{n-1}$ cosec^n-2 x cot x + $\dfrac{n-2}{n-1}$ I_n-2