高校生のための数学の質問スレPART251

積分∫{sin(x)}^n{cos(x)}^3 dx
が解答と違うのですが自分でやった
部分積分の結果（上）の何処が違うのか教えて
頂けませんでしょうか。お願いします。
=∫[{sin(x)}^(n)*cos(x)]*{cos(x)}^2
=[(1/(n+1))*{sin(x)}^(n+1)]*{cos(x)}^2
-∫[(1/(n+1))*{sin(x)}^(n+1)]*2cos(x)*{-sin(x)} dx
=[(1/(n+1))*{sin(x)}^(n+1)]*{cos(x)}^2
+∫[(1/(n+1))*{sin(x)}^(n+2)]*2cos(x)dx
=[(1/(n+1))*{sin(x)}^(n+1)]*{cos(x)}^2
+[(1/(n+1))*(1/(n+3))*{sin(x)}^(n+3)]*2
解答
=∫[{sin(x)}^(n)*cos(x)]*{cos(x)}^2dx
=∫[{sin(x)}^(n)*cos(x)]*[1-{sin(x)}^2]dx
=∫[{sin(x)}^(n)-{sin(x)}^(n+2)]*cos(x)dx
=(1/(n+1))*{sin(x)}^(n+1)-(1/(n+3))*{sin(x)}^(n+3)