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

解

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

- \frac{cos ( 2 x )}{x} - 2 Si (2 x) + C

解答ステップ

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

u置換積分法を適用する

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

定数を除く:

\int a \cdot f (x) d x = a \cdot \int f (x) d x

= 2 \cdot \int \frac{cos ( u )}{u ^{2}} d u

部分積分を適用する

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

\int \frac{sin ( u )}{u} d u = Si (u)

= 2 (- \frac{cos ( u )}{u} - Si (u))

代用を戻す

u = 2 x

= 2 (- \frac{cos ( 2 x )}{2 x} - Si (2 x))

簡素化

2 (- \frac{cos ( 2 x )}{2 x} - Si (2 x)) : - \frac{cos ( 2 x )}{x} - 2 Si (2 x)

= - \frac{cos ( 2 x )}{x} - 2 Si (2 x)

定数を解答に追加する

= - \frac{cos ( 2 x )}{x} - 2 Si (2 x) + C