integral of (sin(2x)-2)/(cos^2(x))

解

\int \frac{sin ( 2 x ) - 2}{cos ^{2} ( x )} d x

- ln (cos^{2} (x)) - 2 tan (x) + C

解答ステップ

\int \frac{sin ( 2 x ) - 2}{cos ^{2} ( x )} d x

拡張

\frac{sin ( 2 x ) - 2}{cos ^{2} ( x )} : \frac{sin ( 2 x )}{cos ^{2} ( x )} - \frac{2}{cos ^{2} ( x )}

= \int \frac{sin ( 2 x )}{cos ^{2} ( x )} - \frac{2}{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{sin ( 2 x )}{cos ^{2} ( x )} d x - \int \frac{2}{cos ^{2} ( x )} d x

\int \frac{sin ( 2 x )}{cos ^{2} ( x )} d x = - ln (cos^{2} (x))

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

= - ln (cos^{2} (x)) - 2 tan (x)

定数を解答に追加する

= - ln (cos^{2} (x)) - 2 tan (x) + C