お勉強しよう 〕 数学.積分

2016/9-2011 Yuji.W

部分積分

. 部分積分 指数関数 対数関数 三角関数 ※ 積分定数は省略 .

◇ ベクトル<A> 単位ベクトル<Au> 内積* 外積# 〔物理定数
◆ ネイピア数 e 虚数単位 i exp(i*x)=expi(x) 微分;x 積分$ 10^x=Ten(x)

部分積分

◆ 2つの関数 f(x) , g(x) 積 f*g の積分を求めたい

ただし ${f*dx}=F(x) g;x は簡単に求める事ができるとする

■ (F*g);x=(F;x)*g+F*(g;x)

積分して F*g=${(F;x)*g*dx}+${F*(g;x)*dx}

 ${f*g*dx}=${(F;x)*g*dx}=F*g-${F*(g;x)*dx} .

☆指数関数への利用☆

■ ${x*exp(x)*dx}
=${x*[exp(x);x]*dx}
=x*exp(x)-${1*exp(x)}
=x*exp(x)-exp(x)

≫ ${x*exp(x)*dx}=x*exp(x)-exp(x) .

■ ${x*exp(a*x)*dx}
=(1/a^2)*${(a*x)*exp(a*x)*d(a*x)}
=(1/a^2)*[(a*x)*exp(a*x)-exp(a*x)]
=(1/a)*x*exp(a*x)-(1/a^2)*exp(a*x)

≫ ${x*exp(a*x)*dx}=(1/a)*x*exp(a*x)-(1/a^2)*exp(a*x) .

■ ${x^2*exp(x)*dx}
=${x^2*[exp(x);x]*dx}
=x^2*exp(x)-2*${x*exp(x)*dx}
=x^2*exp(x)-2*[x*exp(x)-exp(x)]
=(x^2-2*x+2)*exp(x)

≫ ${x^2*exp(x)*dx}=(x^2-2*x+2)*exp(x) .
■ ${x^2*exp(a*x)*dx}
=(1/a^3)*${(a*x)^2*exp(a*x)*d(a*x)}
=(1/a^3)*[(a*x)^2-2*a*x+2]*exp(a*x)
=[(1/a)*x^2-(2/a^2)*x+2/a^3]*exp(a*x)

≫ ${x^2*exp(a*x)*dx}=[(1/a)*x^2-(2/a^2)*x+2/a^3]*exp(a*x) .

 ${x*exp(x)*dx}=x*exp(x)-exp(x)

 ${x*exp(a*x)*dx}=(1/a)*x*exp(a*x)-(1/a^2)*exp(a*x)

 ${x^2*exp(x)*dx}=(x^2-2*x+2)*exp(x)

 ${x^2*exp(a*x)*dx}=[(1/a)*x^2-(2/a^2)*x+2/a^3]*exp(a*x)

◇対数関数への利用◇

■ ${ln|x|*dx}
=${(x;x)*ln|x|*dx}
=x*ln|x|-${x*(1/x)*dx}
=x*ln|x|-${1*dx}
=x*ln|x|-x
=x*(ln|x|-1)

≫ ${ln|x|*dx}=x*(ln|x|-1) .

■ ${ln|a*x|*dx}
=(1/a)*${ln|a*x|*d(a*x)}
=(1/a)*[(a*x)*ln|a*x|-a*x]
=x*ln|a*x|-x
=x*(ln|a*x|-1)

≫ ${ln|a*x|*dx}=x*(ln|a*x|-1) .

■ ${x*ln|x|*dx}
=${[(x^2/2);x]*ln|x|*dx}
=(x^2/2)*ln|x|-${[(x^2/2)]/x*dx}
=(x^2/2)*ln|x|-${(x/2)*dx}
=(1/2)*x^2*ln|x|-(1/4)*x^2
=(x^2/4)*(2*ln|x|-1)

≫ ${x*ln|x|*dx}=(x^2/4)*(2*ln|x|-1) .

★ ${(2x+3)*ln|x|*dx}
=2*${x*ln|x|*dx}+3*${ln|x|*dx}
=2*(x^2/4)*(2*ln|x|-1)+3*x*(ln|x|-1)
=x*(x+3)*ln|x|-x*(x+6)/2

 ${ln|x|*dx}=x*(ln|x|-1)

 ${ln|a*x|*dx}=x*(ln|a*x|-1)

 ${x*ln|x|*dx}=(x^2/4)*(2*ln|x|-1)

☆三角関数への利用☆

■ ${x*sin(x)*dx}
=${x*[-cos(x);x]*dx}
=-x*cos(x)+${cos(x)*dx}
=-x*cos(x)+sin(x)

≫ ${x*cos(x)*dx}=x*sin(x)+cos(x) .

■ ${x*sin(a*x)*dx}
=${x*[-cos(a*x);x/a]*dx}
=-x*cos(a*x)/a+${cos(a*x)*dx}/a
=-x*cos(a*x)/a+sin(a*x)/a^2
.

 ${x*cos(a*x)*dx}=x*sin(a*x)/a+cos(a*x)/a^2 .

■ ${x^2*cos(x)*dx}
=${x^2*[sin(x);x]*dx}
=x^2*sin(x)-2*${x*sin(x)dx}
=x^2*sin(x)-2*[-x*cos(x)+sin(x)]
=(x^2-2)*sin(x)+2*x*cos(x)
.

 ${x^2*sin(x)*dx}=(-x^2+2)*cos(x)+2*x*sin(x) .

■ ${x^2*cos(a*x)*dx}
=${x^2*[sin(a*x);x/a]*dx}
=x^2*sin(a*x)/a-2*${x*sin(a*x)dx}/a
=x^2*sin(a*x)/a-2*[-x*cos(a*x)/a+sin(a*x)/a^2]/a
=(x^2/a-2/a^3)*sin(a*x)+2*x*cos(a*x)/a^2
.

{確かめ} 右辺;x
=2*x*sin(a*x)/a+(x^2-2/a^2)*cos(a*x)+2*cos(a*x)/a^2-2*x*sin(a*x)/a
=x^2*cos(a*x)

 ${x^2*sin(a*x)*dx}=(-x^2/a+2/a^3)*cos(a*x)+2*x*sin(a*x)/a^2 .

 ${x*cos(x)*dx}=x*sin(x)+cos(x)

 ${x*cos(a*x)*dx}=x*sin(a*x)/a+cos(a*x)/a^2

 ${x*sin(a*x)*dx}=-x*cos(a*x)/a+sin(a*x)/a^2

 ${x^2*cos(x)*dx}=(x^2-2)*sin(x)+2*x*cos(x)

 ${x^2*cos(a*x)*dx}=(x^2/a-2/a^3)*sin(a*x)+2*x*cos(a*x)/a^2

 ${x^2*sin(x)*dx}=(-x^2+2)*cos(x)+2*x*sin(x)

 ${x^2*sin(a*x)*dx}=(-x^2/a+2/a^3)*cos(a*x)+2*x*sin(a*x)/a^2

◇三角関数*対数関数 への利用-2-◇

■ I=${cos(x)*exp(x)*dx}

 I=${(sin(x);x)*exp(x)*dx}=sin(x)*exp(x)-${sin(x)*exp(x)*dx}

ここで ${sin(x)*exp(x)*dx}
=${(-cos(x);x)*exp(x)*dx}
=-cos(x)*exp(x)+${cos(x)*exp(x)*dx}
=-cos(x)*exp(x)+I だから、

 I=sin(x)*exp(x)+cos(x)*exp(x)-I

 I=(1/2)*exp(x)*(cos(x)+sin(x))

 ${cos(x)*exp(x)*dx}=(1/2)*exp(x)*(cos(x)+sin(x)) .

■ I=${cos(a*x)*exp(b*x)*dx}

 I
=${{[(1/a)*sin(a*x)];x}*exp(b*x)*dx}
=(1/a)*sin(a*x)*exp(b*x)-(b/a)*${sin(a*x)*exp(b*x)*dx}

ここで ${sin(a*x)*exp(b*x)*dx}
=${{[-(1/a)*cos(a*x)];x}*exp(b*x)*dx}
=-(1/a)*cos(a*x)*exp(b*x)+(b/a)*${cos(a*x)*exp(b*x)*dx}
=-(1/a)*cos(a*x)*exp(b*x)+(b/a)*I だから、

 I
=(1/a)*sin(a*x)*exp(b*x)-(b/a)*[-(1/a)*cos(a*x)*exp(b*x)+(b/a)*I]
=(1/a)*sin(a*x)*exp(b*x)+(b/a^2)*cos(a*x)*exp(b*x)-(b/a)^2*I

 I=exp(b*x)*[a*sin(a*x)+b*cos(a*x)]/(a^2+b^2)

 ${cos(a*x)*exp(b*x)*dx}
=exp(b*x)*[a*sin(a*x)+b*cos(a*x)]/(a^2+b^2) 
.

 ${cos(x)*exp(x)*dx}=(1/2)*exp(x)*(cos(x)+sin(x))

 ${cos(a*x)*exp(b*x)*dx}
=exp(b*x)*[a*sin(a*x)+b*cos(a*x)]/(a^2+b^2)

.  部分積分  .

inserted by FC2 system