integral of (pisec(x)-tan(x))sec(x)

解

\int (π sec (x) - tan (x)) sec (x) d x

π tan (x) - sec (x) + C

解答ステップ

\int (π sec (x) - tan (x)) sec (x) d x

三角関数の公式を使用して書き換える

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

拡張

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

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

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

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

= π tan (x) - sec (x)

定数を解答に追加する

= π tan (x) - sec (x) + C