Question

$\int \frac{co{\sec }^{2}x-203}{{\left(\cos x\right)}^{203}}dx$

• A. $\frac{-\cot x}{{\left(\cos x\right)}^{203}}$
• B. $\frac{\cos x}{{\left(co\sec x\right)}^{203}}$
• C. $\frac{-\tan x}{{\left(co\sec x\right)}^{203}}$
• D. $\frac{\cot x}{{\left(\sin x\right)}^{203}}$

Answer 1

The correct option is A $\frac{-\cot x}{{\left(\cos x\right)}^{203}}$

$\int \left({\cos }^{-203}x\right)co{\sec }^{2}xdx-\int \frac{203}{{\left(\cos x\right)}^{203}}dx$
$=I_1-I_2$
Integrate $I_1$ by parts
$\therefore I_1=\left({\cos }^{-203}x\right)(-\cot x)-\int \cot x.203{\cos }^{-204}x(-\sin x)dx$
$I_1=-\frac{\cot x}{{\left(\cos x\right)}^{203}}+\int \frac{203}{{\left(\cos x\right)}^{203}}dx+C$
$\therefore I_1-I_2=-\frac{\cot x}{{\left(\cos x\right)}^{203}}+C$