ex21-22-23.mw

Exercice 21

> Int(cos(t)^4*sin(t)^5,t)=int(cos(t)^4*sin(t)^5,t);

Int(cos(t)^4*sin(t)^5, t) = -1/9*cos(t)^5*sin(t)^4-4/63*sin(t)^2*cos(t)^5-8/315*cos(t)^5

> Int(cos(t)^3*sin(t)^4,t)=int(cos(t)^3*sin(t)^4,t);

Int(cos(t)^3*sin(t)^4, t) = -1/7*cos(t)^4*sin(t)^3-3/35*sin(t)*cos(t)^4+1/35*cos(t)^2*sin(t)+2/35*sin(t)

> Int(cos(t)^6,t)=int(cos(t)^6,t);

Int(cos(t)^6, t) = 1/6*cos(t)^5*sin(t)+5/24*cos(t)^3*sin(t)+5/16*cos(t)*sin(t)+5/16*t

> Int(cos(t)^4*sin(t)^2,t)=int(cos(t)^4*sin(t)^2,t);

Int(cos(t)^4*sin(t)^2, t) = -1/6*cos(t)^5*sin(t)+1/24*cos(t)^3*sin(t)+1/16*cos(t)*sin(t)+1/16*t

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> Exercice 22

> Int((t+1)/(t^2+t+1)^2,t)=int((t+1)/(t^2+t+1)^2,t);

Int((t+1)/(t^2+t+1)^2, t) = 1/3*(t-1)/(t^2+t+1)+2/9*3^(1/2)*arctan(1/3*(2*t+1)*3^(1/2))

> Int(1/(t^3+1),t)=int(1/(t^3+1),t);

Int(1/(t^3+1), t) = 1/3*ln(t+1)-1/6*ln(t^2-t+1)+1/3*3^(1/2)*arctan(1/3*(2*t-1)*3^(1/2))

> Int(1/(t^4+t^2+1),t)=int(1/(t^4+t^2+1),t);

Int(1/(t^4+t^2+1), t) = -1/4*ln(t^2-t+1)+1/6*3^(1/2)*arctan(1/3*(2*t-1)*3^(1/2))+1/4*ln(t^2+t+1)+1/6*3^(1/2)*arctan(1/3*(2*t+1)*3^(1/2))

> Int((t^2-1)^2/(t*(t^2+1)),t)=int((t^2-1)^2/(t*(t^2+1)),t);

Int((t^2-1)^2/(t*(t^2+1)), t) = 1/2*t^2-2*ln(t^2+1)+ln(t)

> Int((1-t)/(t^2+t+1)^2,t)=int((1-t)/(t^2+t+1)^2,t);

Int((1-t)/(t^2+t+1)^2, t) = 1/3*(3*t+3)/(t^2+t+1)+2/3*3^(1/2)*arctan(1/3*(2*t+1)*3^(1/2))

> Int((t^3-3*t+1)/((t-1)*(t^2+1)^2),t)=int((t^3-3*t+1)/((t-1)*(t^2+1)^2),t);

Int((t^3-3*t+1)/((t-1)*(t^2+1)^2), t) = -1/4*ln(t-1)+1/8*ln(t^2+1)+1/8*(-10*t-6)/(t^2+1)

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> Exercice 23

> Int(1/(sin(t)*(sin(t)-cos(t))),t)=int(1/(sin(t)*(sin(t)-cos(t))),t);

Int(1/(sin(t)*(sin(t)-cos(t))), t) = ln(tan(t)-1)-ln(tan(t))

> Int(1/cos(t),t)=int(1/cos(t),t);

Int(1/cos(t), t) = ln(sec(t)+tan(t))

> Int(1/(sin(t)^4+cos(t)^4),t)=int(1/(sin(t)^4+cos(t)^4),t);

Int(1/(sin(t)^4+cos(t)^4), t) = 1/8*2^(1/2)*ln((tan(t)^2+tan(t)*2^(1/2)+1)/(tan(t)^2-tan(t)*2^(1/2)+1))+1/2*2^(1/2)*arctan(tan(t)*2^(1/2)+1)+1/2*2^(1/2)*arctan(tan(t)*2^(1/2)-1)+1/8*2^(1/2)*ln((tan(t)...

> Int(1/(cos(t)*cos(2*t)),t)=int(1/(cos(t)*cos(2*t)),t);

Int(1/(cos(t)*cos(2*t)), t) = -1/2*ln(1+sin(t))+1/2*ln(sin(t)-1)+2^(1/2)*arctanh(sin(t)*2^(1/2))

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