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数学 高校生

書き込みで見づらくて申し訳ありません。 (2)での変形がなかなか思いつかないのですが、コツなどあれば教えて頂きたいです。 自分で思ったのはcos^n-2x を登場させるためにcos^n-1xを部分積分の微分側にして、結果としてsin^m+2xが出てきてしまうけれど、sin^... 続きを読む

重要 例題237定積分と漸化式 (2) OOOO0 nを0以上の整数として, Im.n=\ sin"xcos"xdxとする。 m。 0 |次の等式を証明せよ。ただし, sin°x=cos°x=1 である。 (1) Im.n=In,m (2) Im.n- n-1 夜-4(n22) m+n p.390 基本事項 2, 重要218,236 針>(1) sin ーx=COSx, COs 2 cos(ー -x)=sinx [sinと cos が入れ替わる] に注目し, x=-tとおき換えて計算し,後で変数さをxに直す。 (2) sin"xcos"x=(sin"xcos.x)cos-1x として 部分積分法 を用いる。 更に、sin"+2xcos"-2x=sin"xcos"-?x- Isin"xcos"x から 同形出現。 ガー2 解答 0 x=ー --tとおくと dx=-dt 元 xとtの対応は右のようになる。 2 x 元 よって Im= sin"xcos"xdx t 0 2 0 cos"(-):(-1)dt=\} sin":xcos":xdx==/.m 2) n22のとき Ssin* 4ぎは確使 からやる x cos"x dx=\(sin"xcosx)cos" sinm+1x COS ) os xde ー1xdx= 、-1xdx m+1 (sin"+1xcos"ー!x Ssinm *(n-1)cos"-2x(-sinx)dx #ー m+1 sin+1 x COS m+1 -1文 カ-1 群+2 m+と xCos da の また Ssin*xcos""xdx=Ssin"xcos"-?a{-coss -Jsin"s x dx= 加+2 一2 XCOS -cos"x) -Jein"xcosx dx-\sin"xcos"xdx の の, のから (sin"xcos"xdx= Sinm*!xcos"-!x Ssin" n-1 m+n x cos"x dx= sin"xcos"-? xdx m+n 「sin"+1xcos"-1 slx1f+ カS sin"xcos"="xdx ゆえに sin"x cos"xdx= n-1( m+n Jo n-1Imn-2 したがって Im= m+n

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英語 高校生

答えと解説お願いします!!

子習日 above a nice restaurant. The student ate only once a day before his evening studies (10) )A A Wise Judgment 名判決 Vdados just 1 Many years ago in Japan, a hungry student lived 1n a small room that woo having ーa small *?rice ball. However, . he did not feel he was alone when he was 1) meals in his room. He could smell the delicious food from the restaurant, and b。 restaurant with many could also imagine that he was eating the delicious food at the 5 people he never met. He really enjoyed his dinner time. 2 One day the restaurant owner came to learnwhat the student was doing, and 2) so he asked him for some money. The owner did not want him to smell the food from the restaurant for free. But the student was so poor that he did not have much money at all. So the owner called the police and asked them to *3 °put the student into 10 prison. 3 Soon there was a *4 "court case. *Judge Nomura was very famous because he was so clever. The owner explained the problem with the student. He explained how much money the restaurant needed for all the people who worked there. The student 3) started to worry because he was so poor. Judge Nomura thought for a minute and 15 then began to talk. 4 “It is clear that you must give the owner some money for received from the restaurant," he said to the student. "Do you have any money with what you have 4) vOu2" The restaurant owner smiled. The student found a few coins in his bag and 20 took them out. The judge told him to 6throw them up and catch them. The student * looked sad, but did so. 5) *7made a small sound. “Good!" said Judge The coins Nomura. “The sound of money is enough for the smell of food." I 300 wordsI

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