Maths · Integration by substitution, by parts and by partial fractions
^{99}(\cos x)^{-101} d x=_{-} \ldots-C_{ } \)
\( \int(\sin x)^{99}(\cos x)^{-101} d x=_{-} \ldots-C_{ } \)
- A. \( \frac{(\tan x)^{100}}{100} \)
- B. \( \frac{(\tan x)^{2}}{2} \)
- C. \( \frac{(\tan x)^{98}}{98} \)
- D. \( \frac{(\tan x)^{97}}{97} \)
Step-by-step solution
Rewrite the integrand as \(\tan^{99} x \sec^2 x\). Substitute \(u = \tan x\), \(du = \sec^2 x\,dx\), giving \(\int u^{99} du = \frac{u^{100}}{100} + C = \frac{(\tan x)^{100}}{100} + C\).