Hola

I1 = ∫ r^2 dr/(4 - r^2)^3/2

r = 2 sen(x)

r^2 = 4 sen^2(x)

4 - r^2 = 4 - 4 sen^2(x) = 4 cos^2(x)

(4 - r^2)^(3/2) = (4 cos^2(x))^(3/2) = (2 cos(x))^3 = 8 cos^3(x)

dr = 2 cos(x)

además

2 sen(x) = r

2 cos(x) = (4 - r^2)^(1/2)

tan(x) = r/(4 - r^2)^(1/2)

I1 = ∫ 4 sen^2(x) (2 cos(x) dx)/(8 cos^3(x))

I1 = ∫ (sen^2(x) dx)/(cos^2(x))

I1 = ∫ tan^2(x) dx

I1 = ∫ (sec^2(x) - 1) dx

I1 = ∫ sec^2(x) dx - ∫dx

I1 = tan(x) - x + C

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I1 = ( r/(4 - r^2)^(1/2) ) - asen(r/2) + C

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I2 = ∫√(x^2+16) dx / x

x = 4 tan(u)

x^2 + 16 = 16 (1 + tan^2(u)) = 16 sec^2(u)

√(x^2+16) = 4 sec(u)

dx = 4 sec^2(u) du

I2 = ∫ 4 sec(u) ( 4 sec^2(u) du)/( 4 tan(u))

I2 = 4 ∫ du/(sen(u) cos^2(u))

I2 = 4 ∫ sen(u) du/(sen^2(u) cos^2(u))

w = cos(u)

dw = - sen(u) du

I2 = (-4) ∫ dw/( (1 - w^2) w^2 )

I2 = (-4) ∫ ( (1 - w^2) + w^2 )dw/( (1 - w^2) w^2 )

I2 = (-4) ∫ (dw / w^2) - (4)∫ (dw /(1- w^2))

I2 = (-4) ∫ (dw / w^2) - (4)∫ (dw /(1- w)(1+w))

I2 = (-4) ∫ (dw / w^2) - (2)∫ ((1- w)+(1+w))dw /(1- w)(1+w)

I2 = (-4) ∫ (w^(-2) dw ) - (2)∫ dw /(1- w) - (2)∫ dw /(1+ w)

I2 = (-4)/(-1) w^(-1) + 2 ln(1 - w) - 2 ln(1 + w) +C

Ahora

w = cos(u)

I2 = 4 sec(u) + 2 ln( (1 - cos(u))/(1 + cos(u)) ) + C

I2 = 4 sec(u) + 2 ln( (sec(u) - 1)/(sec(u) + 1) ) + C

ahora

x = 4 tan(u)

√(x^2+16) = 4 sec(u)

I2 = 4 √(x^2+16) + 2 ln( (√(x^2+16) - 1)/(√(x^2+16) + 1) ) + C

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