탄젠트 거듭제곱 적분

ㄱ. $\displaystyle a _{1} + a _{3} = \int _{0} ^{\frac{\pi}{4}} {} \tan x dx + \int _{0} ^{\frac{\pi}{4}} {} \tan ^{3} x dx$ $\displaystyle = \int _{0} ^{\frac{\pi}{4}} {} \tan x ( 1 + \tan ^{2} x ) dx$$\displaystyle = \int _{0} ^{\frac{\pi}{4}} {} \tan x \sec ^{2} x dx$ $\tan x = t$일 때, $\displaystyle \sec ^{2} x = \frac{dt}{dx}$이므로 $\displaystyle \int _{0} ^{\frac{\pi}{4}} {} \tan x \sec ^{2} x dx = \int _{0} ^{1} {} t dt$ $\displaystyle = \left[ \frac{1}{2} t ^{2} \right] _{0} ^{1} = \frac{1}{2}$ (참) ㄴ. 마찬가지로 생각하면 $a _{2} + a _{4}$$\displaystyle = \int _{0} ^{1} {} t ^{2} dt$$\displaystyle = \left[ \frac{1}{3} t ^{3} \right] _{0} ^{1} = \frac{1}{3}$ ∴ $\displaystyle a _{1} + a _{2} + a _{3} + a _{4} = \frac{1}{2} + \frac{1}{3}$(참) ㄷ. 일반적으로 $a _{4 k + 1} + a _{4 k + 2} + a _{4 k + 3} + a _{4 k + 4}$ $= ( a _{4 k + 1} + a _{4 k + 3} ) + ( a _{4 k + 2} + a _{4 k + 4} )$ $\displaystyle = \int _{0} ^{1} {} t ^{4 k + 1} dt$$\displaystyle + \int _{0} ^{1} {} t ^{4 k + 2} dt$$\displaystyle = \frac{1}{4 k + 2} + \frac{1}{4 k + 3}$ ∴ $\displaystyle \sum\limits _{k = 1} ^{100} a _{k} =$$\displaystyle \sum\limits _{k = 0} ^{24} ( a _{4 k + 1} + a _{4 k + 2} + a _{4 k + 3} + a _{4 k + 4} )$ $\displaystyle = \sum\limits _{k = 0} ^{24} \left( \frac{1}{4 k + 2} + \frac{1}{4 k + 3} \right)$ $\displaystyle = \frac{1}{2} + \frac{1}{3} + \frac{1}{6} + \frac{1}{7} + \cdots + \frac{1}{98} + \frac{1}{99}$ (거짓)