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== 定積分の部分積分法 ==

○不定積分に関する部分積分の公式

から定積分に関する部分積分の公式

が導かれる.
(詳細)
 積の微分法の公式

の両辺を区間a≦x≦bにおいて積分すると
 左辺のの1つの原始関数はだから
 左辺はになる.
 右辺は

したがって

移項すると,定積分に関する部分積分の公式が得られる.

f' (x)g(x)f(x)g'(x)はどちらが左辺でどちらが右辺でもよい.すなわち,次の式でもよい.

【例】

の値を求めてください.
(解答)
f(x)=x微分する→f '(x)=1
g(x)=ex←積分するg'(x)=ex
x=f(x)
ex=g'(x)
とおいて,左辺が

になっているものとして公式を適用する.(左辺は表の水色背景色の組だとする)
 表のように,あらかじめf(x)を微分したものf '(x)
及びg'(x)を積分したものg(x)も求めておく.

に当てはめると

…(答)


【問題】 次の空欄を埋めなさい。
空欄にはスペースを使わずに半角の「アルファベット小文字または数字」だけを使用するものとします.
(1)

π20wwwx sinx dx
◇考え方◇ sinx だけならば積分が求められるが x が掛けてあるので求められない
→ x を微分する側:現在 f の側とする.

( )は左辺の初めの形

下げる:( f ) = x → f ' = 1 
    g =−cosx ← (g') = sinx
:上げる




π20wwwxsinx dx=[ncosx+π20wwwdx

= (0−0) +(1−0) = 1

(2)

e1wwwx log x dx
◇考え方◇ log x の積分が求められない
log xを微分する側:現在 f の側とする.

( )は左辺の初めの形

下げる:( f ) = log x → f' = .1xn 

    g =
.x22nn(g') = x:上げる



e1wwwxlogx dx=[n.x22nne1www.x22nn.1nnnndx

= (.e22n−0)−e1www.x2ndx = .e22n[n.x24n

=.e2+1nnnn
(3)

e1wwwlog x dx
◇考え方◇ log x の積分が求められない
log x と考えて,log x を微分する側:現在 f の側とする.

( )は左辺の初めの形

下げる:( f ) = log x → f' = .1xn

    g = x ← (g') = 1
:上げる



e1wwwlogx dx= [n x e1www x .1xn dx

=(e−0)− [n =e−(e−1)=1


(4)

21www(x−1)(x−2)2 dx
◇考え方◇  wn(x−2)2 dxだけならば求められる.

(置換積分を用いた公式): wn(x−2)2dx=.(x−2)33nnnnnn+C

x−1 を微分する側:現在 f の側とする.

( )は左辺の初めの形

下げる:( f ) = x−1 → f' = 1

    g =
.(x−2)33nnnnn(g') = (x−2)2:上げる



21www(x−1)(x−2)2dx=[n(x−1).(x−2)33nnnnn21www.(x−2)3nnnnndx

=(0−0)−[n.(x−2)4nnnnn=.112nn


(5)

π0wwwex sin x dx
◇考え方◇ 直接的には値が求まらないとき,= Iとおいて I の方程式を解く.
 I = (···)−I → I = .(・・・)2nnnn

 I = (・・・) + I型だけは不可.それ以外は,I = (・・・) ± kI 型になればすべて求まる.
※ 1回の部分積分で自分自身 I が登場しなければ,部分積分を2回行う.
(ただし,ある関数を上げて下げると何も残らないので注意.上げるなら2回上げること.)
( )は左辺の初めの形

下げる:( f ) = sin x → f' = cos x
    g = ex(g') = ex
:上げる


π0wwwexsinx dx=[nex π0wwwex dx

=0−π0wwwexcosx dx



( )は左辺の初めの形

下げる:(p) = cos x → p' =−sin x

    q = ex(q') = ex
:上げる


=−([nex cos x + π0wwwex sin x dx )

=−(−eπ)−π0wwwex sin x dx
I = eπ + 1−I

I = .eπ+12nnnn
(6)

e1www(log x)2 dx
◇考え方◇ (3)と同様に(log x)2 を微分する側に選ぶ.


( )は左辺の初めの形

下げる:( f ) = (log x)2 → f' = 2 (log x) .1xn

    g = x ← (g') = 1:上げる


e1www(log x)2 dx

=
[n (log x)2 e1www log x dx = (*)


( )は左辺の初めの形

下げる:(p) = (log x) → p' = .1xn

    q = x ← (q') = 1:上げる


(*) = e−2( [n x log x e1www dx )

= e−2(
[n x ) = e−2(e−e + 1) = e−2


[注]直前にPC版から入られた場合は,自動転送でスマホ版に来ていますので,ブラウザの[戻るキー]では戻れません(堂々巡りになる).下記のリンクを使ってメニューに戻ってください.
...(携帯版)メニューに戻る

...(PC版)メニューに戻る

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