integral of ((1-tan(x))^5)/(cos^2(x))

解

\int \frac{( 1 - tan ( x ) ) ^{5}}{cos ^{2} ( x )} d x

tan (x) - \frac{5}{2} tan^{2} (x) + \frac{10}{3} tan^{3} (x) - \frac{5}{2} tan^{4} (x) + tan^{5} (x) - \frac{tan ^{6} ( x )}{6} + C

解答ステップ

\int \frac{( 1 - tan ( x ) ) ^{5}}{cos ^{2} ( x )} d x

拡張

\frac{( 1 - tan ( x ) ) ^{5}}{cos ^{2} ( x )} : \frac{1}{cos ^{2} ( x )} - \frac{5 tan ( x )}{cos ^{2} ( x )} + \frac{10 tan ^{2} ( x )}{cos ^{2} ( x )} - \frac{10 tan ^{3} ( x )}{cos ^{2} ( x )} + \frac{5 tan ^{4} ( x )}{cos ^{2} ( x )} - \frac{tan ^{5} ( x )}{cos ^{2} ( x )}

= \int \frac{1}{cos ^{2} ( x )} - \frac{5 tan ( x )}{cos ^{2} ( x )} + \frac{10 tan ^{2} ( x )}{cos ^{2} ( x )} - \frac{10 tan ^{3} ( x )}{cos ^{2} ( x )} + \frac{5 tan ^{4} ( x )}{cos ^{2} ( x )} - \frac{tan ^{5} ( x )}{cos ^{2} ( x )} d x

総和規則を適用する:

\int f (x) \pm g (x) d x = \int f (x) d x \pm \int g (x) d x

= \int \frac{1}{cos ^{2} ( x )} d x - \int \frac{5 tan ( x )}{cos ^{2} ( x )} d x + \int \frac{10 tan ^{2} ( x )}{cos ^{2} ( x )} d x - \int \frac{10 tan ^{3} ( x )}{cos ^{2} ( x )} d x + \int \frac{5 tan ^{4} ( x )}{cos ^{2} ( x )} d x - \int \frac{tan ^{5} ( x )}{cos ^{2} ( x )} d x

\int \frac{1}{cos ^{2} ( x )} d x = tan (x)

\int \frac{5 tan ( x )}{cos ^{2} ( x )} d x = \frac{5}{2} tan^{2} (x)

\int \frac{10 tan ^{2} ( x )}{cos ^{2} ( x )} d x = \frac{10}{3} tan^{3} (x)

\int \frac{10 tan ^{3} ( x )}{cos ^{2} ( x )} d x = \frac{5}{2} tan^{4} (x)

\int \frac{5 tan ^{4} ( x )}{cos ^{2} ( x )} d x = tan^{5} (x)

\int \frac{tan ^{5} ( x )}{cos ^{2} ( x )} d x = \frac{tan ^{6} ( x )}{6}

= tan (x) - \frac{5}{2} tan^{2} (x) + \frac{10}{3} tan^{3} (x) - \frac{5}{2} tan^{4} (x) + tan^{5} (x) - \frac{tan ^{6} ( x )}{6}

定数を解答に追加する

= tan (x) - \frac{5}{2} tan^{2} (x) + \frac{10}{3} tan^{3} (x) - \frac{5}{2} tan^{4} (x) + tan^{5} (x) - \frac{tan ^{6} ( x )}{6} + C