数学 積分 2018/5 Yuji.W
☆ 積分
sin(a)/[A-B*cos(a)]^(3/2)
sin(a)*cos(a)/[A-B*cos(a)]^(3/2) ☆
◎ 積分
◇ ベクトル <A> 内積 * 外積 # 10^x=Ten(x) 微分
;x 時間微分
' 積分 $
ネイピア数 e e^x=exp(x) i^2=-1 e^(i*x)=exp(i*x)=expi(x) 00
〓 関数 〓
◆ f(a)=sin(a)/[1-cos(a)]^(3/2) g(a)=sin(a)*cos(a)/[1-cos(a)]^(3/2)
■ f(0)=sin(0)/[1-cos(0)]^(3/2)=0/0 ?
f(Pi/6)
=sin(Pi/6)/[1-cos(Pi/6)]^(3/2)
=0.5/(1-root3/2)^(3/2)
=10.2
f(Pi/3)
=sin(Pi/3)/[1-cos(Pi/3)]^(3/2)
=(root3/2)/0.5^(3/2)
=0.866/0.354
=2.45
f(Pi/2)=sin(Pi/2)/[1-cos(Pi/2)]^(3/2)=1
f(2*Pi/3)
=sin(2*Pi/3)/[1-cos(2*Pi/3)]^(3/2)
=sin(2*Pi/3)/[1-cos(2*Pi/3)]^(3/2)
=0.866/1.837
=0.471
f(Pi)=sin(Pi)/[1-cos(Pi)]^(3/2)=0
S(Pi/2~Pi)=[f(Pi/2)+f(Pi)]*(Pi-Pi/2)/2=(1+0)*Pi/4=0.785
■ g(0)=cos(0)*f(0)=0/0 ?
g(Pi/6)=cos(Pi/6)*f(Pi/6)=(root3/2)*10.2=8.83
g(Pi/3)=cos(Pi/3)*f(Pi/3)=0.5*2.45=1.23
g(Pi/2)=cos(Pi/2)*f(Pi/2)=0*1=0
g(2*Pi/3)=cos(2*Pi/3)*f(2*Pi/3)=(-0.5)*0.471=-0.236
g(Pi)=cos(Pi)*f(0)=1*0=0
S(Pi/3~Pi/2)=[g(Pi/3)+g(Pi/2)]*(Pi/2-Pi/3)/2=1.23*Pi/12=0.32
〓 積分 〓
■ ${dx/root(A-B*x)}=-(2/B)*root(A-B*x)+積分定数
■ ${dx/(A-B*x)^(3/2)}=(2/B)/root(A-B*x)+積分定数
■ ${x*dx/(A-B*x)^(3/2)}=(2/B^2)*(2*A-B*x)/root(A-B*x)+積分定数
〓 微分 〓
■ {root[A-B*cos(a)]};a=(B/2)*sin(a)/root[A-B*cos(a)]
■ {1/root[A-B*cos(a)]};a
=-({root[A-B*cos(a)]};a)/[A-B*cos(a)]
=-(B/2)*sin(a)/[A-B*cos(a)]^(3/2)
■ {cos(a)/root[A-B*cos(a)]};a
=-sin(a)/root[A-B*cos(a)]+cos(a)*{-(B/2)*sin(a)/[A-B*cos(a)]^(3/2)}
=-sin(a)/root[A-B*cos(a)]-(B/2)*sin(a)*cos(a)/[A-B*cos(a)]^(3/2)
=-sin(a)*{[A-B*cos(a)]+(B/2)*cos(a)}/[A-B*cos(a)]^(3/2)
=-(1/2)*sin(a)*[2*A-B*cos(a)]/[A-B*cos(a)]^(3/2)
〓 積分 sin(a)/[A-B*cos(a)]^(3/2) 〓
■ {1/root[A-B*cos(a)]};a=-(B/2)*sin(a)/[A-B*cos(a)]^(3/2)
${sin(a)*da/[A-B*cos(a)]^(3/2)}=-(2/B)/root[A-B*cos(a)]+積分定数 ★_
{別解} cos(a)=t と置くと -sin(a)*da=dt
${sin(a)*da/[A-B*cos(a)]^(3/2)}
=-${dt/(A-B*t)^(3/2)}
=-(2/B)/root(A-B*t)+積分定数
=-(2/B)/root[A-B*cos(a)]+積分定数
{よし!2018/5}
〓 積分 sin(a)*cos(a)/[A-B*cos(a)]^(3/2) 〓
■ {1/root[A-B*cos(a)]};a=-(B/2)*sin(a)/[A-B*cos(a)]^(3/2)
*(-4*A/B^2)
(-4*A/B^2)/root[A-B*cos(a)]};a=+(2*A/B)*sin(a)/[A-B*cos(a)]^(3/2) ①
また {cos(a)/root[A-B*cos(a)]};a
=-(1/2)*sin(a)*[2*A-B*cos(a)]/[A-B*cos(a)]^(3/2)
*(2/B)
(2/B)*{cos(a)/root[A-B*cos(a)]};a
=-sin(a)*[2*A/B-cos(a)]/[A-B*cos(a)]^(3/2) ②
①+② 左辺={[(-4*A/B^2)+(2/B)*cos(a)]/root[A-B*cos(a)]};a
右辺=sin(a)*cos(a)/[A-B*cos(a)]^(3/2)
⇒ sin(a)*cos(a)/[A-B*cos(a)]^(3/2)
={[(-4*A/B^2)+(2/B)*cos(a)]/root[A-B*cos(a)]};a
積分して、
${sin(a)*cos(a)*da/[A-B*cos(a)]^(3/2)}
=[(-4*A/B^2)+(2/B)*cos(a)]/root[A-B*cos(a)]+積分定数
=-(2/B^2)*[2*A-B*cos(a)]/root[A-B*cos(a)]+積分定数
》 ${sin(a)*cos(a)*da/[A-B*cos(a)]^(3/2)}
=-(2/B^2)*[2*A-B*cos(a)]/root[A-B*cos(a)]+積分定数 ★_
{別解} cos(a)=t と置くと -sin(a)*da=dt
${sin(a)*cos(a)*da/[A-B*cos(a)]^(3/2)}
=-${t*dt/(A-B*t)^(3/2)}
=-(2/B^2)*(2*A-B*t)/root(A-B*t)+積分定数
=-(2/B^2)*[2*A-B*cos(a)]/root[A-B*cos(a)]+積分定数
{よし!2018/5}
■ ${sin(a)*da/[A-B*cos(a)]^(3/2)}=-(2/B)/root[A-B*cos(a)]+積分定数
■
${sin(a)*cos(a)*da/[A-B*cos(a)]^(3/2)}
=-(2/B^2)*[2*A-B*cos(a)]/root[A-B*cos(a)]+積分定数
〓 {計算例1} 〓
◆ I(a)=${sin(a)*[2*cos(a)-1]*da/[5-4*cos(a)]^(3/2)}[a:0~a]
I1(a)=${sin(a)*da/[5-4*cos(a)]^(3/2)}[a:0~a]
I2(a)=${sin(a)*cos(a)*da/[5-4*cos(a)]^(3/2)}[a:0~a]
I(a)=I2*2-I1
■ A=5 , B=4
${sin(a)*da/[5-4*cos(a)]^(3/2)}=-(1/2)/root[5-4*cos(a)]+積分定数
I1(a)
={-(1/2)/root[5-4*cos(a)]}[a:0~a]
=(1/2)-(1/2)/root[5-4*cos(a)]
■ ${sin(a)*cos(a)*da/[5-4*cos(a)]^(3/2)}
=-(1/8)*[10-4*cos(a)]/root[5-4*cos(a)]+積分定数
=-(1/4)*[5-2*cos(a)]/root[5-4*cos(a)]+積分定数
I2(a)
={-(1/4)*[5-2*cos(a)]/root[5-4*cos(a)]}[a:0~a]
=3/4-(1/4)*[5-2*cos(a)]/root[5-4*cos(a)]
■ I(a)
=3/2-(1/2)*[5-2*cos(a)]/root[5-4*cos(a)]-(1/2)+(1/2)/root[5-4*cos(a)]
=1-[2-cos(a)]/root[5-4*cos(a)]
》I(a)
=${sin(a)*[2*cos(a)-1]*da/[5-4*cos(a)]^(3/2)}[a:0~a]
=1-[2-cos(a)]/root[5-4*cos(a)] ★_
★ I(Pi)
=${sin(a)*[2*cos(a)-1]*da/[5-4*cos(a)]^(3/2)}[a:0~Pi]
=1-[2-cos(Pi)]/root[5-4*cos(Pi)]
=1-1
=0 ★_
★ I(Pi/3)
=${sin(a)*[2*cos(a)-1]*da/[5-4*cos(a)]^(3/2)}[a:0~Pi/3]
=1-[2-cos(Pi/3)]/root[5-4*cos(Pi/3)]
=1-(3/2)/root3
=1-root3/2
=0.134 ★_
■ ${sin(a)*da/[5-4*cos(a)]^(3/2)}[a:0~a]=1/2-(1/2)/root[5-4*cos(a)]
■
${sin(a)*cos(a)*da/[5-4*cos(a)]^(3/2)}[a:0~a]
=3/4-(1/4)*[5-2*cos(a)]/root[5-4*cos(a)]