integral of (sqrt(1+sin(x)))/(sec(x))

解

\int \frac{1 + sin ( x )}{sec ( x )} d x

sin (x) 1 + sin (x) - \frac{1}{3} (1 + sin (x))^{\frac{3}{2}} + 1 + sin (x) + C

解答ステップ

\int \frac{1 + sin ( x )}{sec ( x )} d x

基本的な三角関数の公式を使用する:

\frac{1}{sec ( x )} = cos (x)

= \int cos (x) 1 + sin (x) d x

部分積分を適用する

= sin (x) 1 + sin (x) - \int \frac{cos ( x ) sin ( x )}{2 1 + sin ( x )} d x

\int \frac{cos ( x ) sin ( x )}{2 1 + sin ( x )} d x = \frac{1}{3} (1 + sin (x))^{\frac{3}{2}} - 1 + sin (x)

= sin (x) 1 + sin (x) - (\frac{1}{3} (1 + sin (x))^{\frac{3}{2}} - 1 + sin (x))

簡素化

= sin (x) 1 + sin (x) - \frac{1}{3} (1 + sin (x))^{\frac{3}{2}} + 1 + sin (x)

定数を解答に追加する

= sin (x) 1 + sin (x) - \frac{1}{3} (1 + sin (x))^{\frac{3}{2}} + 1 + sin (x) + C