First slide

Introduction to integration

Question

$\int \frac{{cosec}^{2} x - 2005}{\cos^{2005} x} dx$

Moderate

A

$\frac{\cot x}{(\cos x)^{2005}} + C$
B

$\frac{\tan x}{(\cos x)^{2005}} +$
C

$- \frac{\tan x}{(\cos x)^{2005}} + C$
D

$\frac{- \cot x}{(\cos x)^{2005}} + C$

Solution

$\begin{array}{l} I = \int (\cos x)^{- 2005} {cosec}^{2} x dx - 2005 \int \frac{dx}{\cos^{2005} x} \\ I = (\cos x)^{- 2005} (- \cot x) - \int (- 2005) (\cos x)^{- 2006} (- \sin x) (- \cot x) dx - 2005 \int \frac{dx}{\cos^{2005} x} \\ I = - \frac{\cot x}{(\cos x)^{2005}} + C \end{array}$

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