Question

Integrate cosec^2x -2005/cos^2005x dx

Answer 1

HELLO DEAR,

GIVEN FUNCTION IS:-$\bold{\int{\frac{cosec^2x - 2005}{cos^{2005}x}}\,dx}$

⇒$\bold{\int{[\frac{cosec^2x}{cos^{2005}x} - \frac{2005}{cos^{2005}x}]}\,dx}$

∴ [integrating by parts]

⇒$\bold{\frac{1}{cos^{2005}x}\int{cosec^2x}\,dx - \int{[d(sec^{2005}x)/dx*\int{cosec^2x}\,dx]}\,dx - \int{\frac{2005}{cos^{2005}x}}\,dx}$

⇒$\bold{\frac{1}{cos^{2005}x}(-cotx) - \int{(2005*sec^{2004}xsecxtanx*(-cotx))}\,dx - \int{\frac{2005}{cos^{2005}x}}\,dx}$​

⇒$\bold{\frac{1}{cos^{2005}x}(-cotx) + \int{\frac{2005}{cos^{2005}x}}\,dx - \int{\frac{2005}{cos^{2005}x}}\,dx}$

⇒$\bold{-\frac{cotx}{cos^{2005}x} + c}$

$\bold{\large{HENCE, \int{\frac{cosec^2x - 2005}{cos^{2005}x}}\,dx = -\sf{\frac{cotx}{cos^{2005}x} + c}}}$

I HOPE ITS HELP YOU DEAR,
THANKS