1.∫(上1下0)t^2 e^2t dt
=0.5∫t^2*d(e^2t)
=0.5[t^2*e^(2t)-∫2t*e^(2t)dt]........部份積分
=0.5[t^2*e^(2t)-∫t*d(e^2t)]
=0.5[t^2*e^(2t)-t*e^(2t)+∫e^(2t)dt]........部份積分
=0.5[t^2*e^(2t)-t*e^(2t)+e^(2t)/2]+k
=(e^(2t)/2)*[t^2-t+1/2]+k
=(e/2)[1-1+1/2]-[0-0+1/2]/2
=e/4-1/4
=(e-1)/4.........ans
2.w1=∫(上x/2下0)e^x sin(x)dx
=sin(x)*e^x-∫cos(x)*e^x*dx........部份積分
=sin(x)*e^x-[e^x*cos(x)+∫e^x*sin(x)dx]........部份積分
=sin(x)*e^x-cos(x)*e^x-w1
2w1=(sin(x)-cos(x))e^x
w1=0.5(sin(x)-cos(x))e^x........ans
=0.5(sin(x/2)-cos(x/2))e^(x/2)-0.5(-cos0)*1
=0.5(sin(x/2)-cos(x/2))e^(x/2)+0.5
=0.5(sin(x/2)-cos(x/2))e^(x/2)+1)........ans
3.w2=∫(上1下0)atan(x)dx
Let y=atan(x) => tan(y)=x, cos(y)=1/√(1+x^2)
=> dx=sec^2(y)dy
w2=∫y*sec^2(y)dy
=∫y*d(tan(y))
=y*tan(y)-∫tan(y)dy........部份積分
=y*tan(y)-r
r=∫tan(y)dy
=∫sin(y)dy/cos(y)
=-∫d(cosy)/cos(y)
=-ln(cos(y))
w2=y*tan(y)+ln(cos(y))
=x*atan(x)+ln(1/√(1+x^2))
=1*atan(1)+ln(1/√2)-0-ln(1)
=pi/4-0.5*ln(2) or 5pi/4-0.5*ln(2)
=0.439 or 3.58........ans
4.∫(上n/2下0)sin^4(x)dx
=∫(1-cos(2x))^2*dx/4
=0.25∫(1-2cos(2x)+cos^2(2x))dx
=0.25∫dx-0.5∫cos(2x)dx+∫(1+cos(4x))dx/2
=x/4-sin(2x)/4+∫dx/2+∫cos(4x)dx/2
=(x-sin(x))/4+x/2+sin(4x)/8
=3x/4-sin(x)/4+sin(4x)/8
=3n/8-sin(n/2)/4+sin(2n)/8-0-0-0
=3n/8-sin(n/2)/4+sin(2n)/8........ans
5.w5=∫sec^3(x)dx
=∫sec(x)*d(tan(x))
=sec(x)*tan(x)-∫tan(x)*sec(x)tan(x)dx........部份積分
=sec(x)*tan(x)-∫tan^2(x)*sec(x)*dx
=sec(x)*tan(x)-∫(sec^2(x)-1)sec(x)dx
=sec(x)*tan(x)-∫sec^3(x)dx+∫sec(x)dx
=sec(x)*tan(x)-w5+∫dx/cos(x)
2*w5=sec(x)*tan(x)+∫cos(x)dx/cos^2(x)=sec(x)*tan(x)+h
h=∫d(sin(x))/(1-sin^2(x))
=∫dy/(1+y)(1-y)......y=sin(x)
=∫dy/2(1+y)+∫dy/2(1-y)
=0.5*[ln(1+y)+ln(1-y)]
=0.5*ln[(1+y)(1-y)]
=0.5*ln(1-y^2)
=0.5*ln(1-sin^2(x))
=0.5*ln(cos^2(x))
=ln(cos(x))
2w5=sec(x)*tan(x)+ln(cos(x))
w5=0.5[sec(x)*tan(x)+ln(cos(x))]+k(積分常數)
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