integral of cos(x)sqrt(sin(x)+1)

解

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

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

解答ステップ

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

部分積分を適用する

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

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

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

簡素化

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

定数を解答に追加する

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