ex24-26-27-28.mw

> Exercice 24

> Int(1/(5-4*cos(t)),t=0..2*Pi)=int(1/(5-4*cos(t)),t=0..2*Pi);

Int(1/(5-4*cos(t)), t = 0 .. 2*Pi) = 2/3*Pi

> Int(1/(2+sin(t)^2),t=0..Pi/6)=int(1/(2+sin(t)^2),t=0..Pi/6);

Int(1/(2+sin(t)^2), t = 0 .. 1/6*Pi) = 1/6*arctan(1/2*2^(1/2))*3^(1/2)*2^(1/2)

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Exercice 26

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

Int(1/cosh(t), t) = 2*arctan(exp(t))

> Int(1/(5*cosh(t)+3*sinh(t)+4),t)=int(1/(5*cosh(t)+3*sinh(t)+4),t);

Int(1/(5*cosh(t)+3*sinh(t)+4), t) = -2/(tanh(1/2*t)+3)

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

Int(1/sinh(t)^3, t) = -1/2*cosh(t)/sinh(t)^2+arctanh(exp(t))

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Exercice 27

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

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

> Int(cos(t)^3/(sin(t)^5),t)=int(cos(t)^3/(sin(t)^5),t);

Int(cos(t)^3/sin(t)^5, t) = -1/4*cos(t)^4/sin(t)^4

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

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

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

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

> Int((t^2+6*t+5)^(3/2),t)=int((t^2+6*t+5)^(3/2),t);

Int((t^2+6*t+5)^(3/2), t) = 1/8*(2*t+6)*(t^2+6*t+5)^(3/2)-3/4*(t^2+6*t+5)^(1/2)*(2*t+6)+6*ln(t+3+(t^2+6*t+5)^(1/2))

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

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

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

Int(t^2*arcsin(t)/(1-t^2)^(1/2), t) = arcsin(t)*(-1/2*(1-t^2)^(1/2)*t+1/2*arcsin(t))-1/4+1/4*t^2-1/4*arcsin(t)^2

> Int((1+t+t^2)/(1+t^2)*exp(arctan(x)),t)=int((1+t+t^2)/(1+t^2)*exp(arctan(x)),t);

Int((1+t+t^2)*exp(arctan(x))/(1+t^2), t) = exp(arctan(x))*t+1/2*exp(arctan(x))*ln(1+t^2)

> Int(1/(cosh(3*t)-cosh(t)),t)=int(1/(cosh(3*t)-cosh(t)),t);

Int(1/(cosh(3*t)-cosh(t)), t) = -1/4/sinh(t)-1/2*arctan(exp(t))

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

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

Exercice 28

> Int(x*arctan(x)^2,x=0..1)=int(x*arctan(x)^2,x=0..1);

Int(x*arctan(x)^2, x = 0 .. 1) = 1/16*Pi^2-1/4*Pi+1/2*ln(2)

> Int(x^(2007)*(x^2+1)^4,x=-1..1)=int(x^(2007)*(x^2+1)^4,x=-1..1);

Int(x^2007*(x^2+1)^4, x = -1 .. 1) = 0

> Int((1+cosh(x))/(1+sinh(x)^2),x=0..1)=int((1+cosh(x))/(1+sinh(x)^2),x=0..1);

Int((1+cosh(x))/(1+sinh(x)^2), x = 0 .. 1) = (-exp(-1)-2*exp(1)*arctan((exp((-1)/2)-exp(1/2))/(exp((-1)/2)+exp(1/2)))+exp(1)-2*exp(-1)*arctan((exp((-1)/2)-exp(1/2))/(exp((-1)/2)+exp(1/2))))/(exp(-1)+e...

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