Integral Calculus

integration

naysa

11 Years agoGrade

1 Answer

jagdish singh singh

9 Years ago

$\hspace{-0.7 cm}$Let $I = \int\frac{\csc^2 x-2005}{\cos^{2005}x}dx = \int \frac{\csc^{2}x}{\cos^{2005}x}dx-\int\frac{2005}{\cos^{2005}}dx$\\\\ Using Integration by parts for $(1)$\\\\ $I = -\frac{\cot x}{\cos^{2005}x}+\int 2005 \cos^{-2006}x\cdot \sin x\cdot \cot xdx-\int\frac{2005}{\cos^{2005}x}dx$\\\\ $I = -\frac{\cot x}{\cos^{2005}x}+\int \frac{2005}{\cos^{2005}x}dx-\int\frac{2005}{\cos^{2005}x}dx$\\\\ $I = \int\frac{\csc^2 x-2005}{\cos^{2005}x}dx = -\frac{\cot x}{\cos^{2005}x}+\mathcal{C}$$