# How do you integrate cos(x)/(4-sin^2(x)) from 0 to 1?

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• Anonymous

sin(x) = 2 tanh(u)

cos(x) dx = 2sech^2(u) du

u = arctanh(sin(x))/2)

∫cos(x) dx /(4-sin^2(x))

= ∫2sech^2(u) du / (4 - 4 tanh^2(u))

= ∫2sech^2(u) du / [(4cosh^2 - 4 sinh^2(u)) / cosh^2(u) )

= ∫2sech^2(u) du / [(4) / cosh^2(u) )

= ∫2sech^2(u) du / [(4) sech^2(u) )

= (1/2) ∫ du = u / 4

= (1/2) arctanh(sin(x))/2)

developing an expression for arctanh

z = arctanh(w)

w = tanh(z) = (e^z - e ^ -z) / (e^z + e^-z) = (e^2z - 1)/ (e^2z + 1)

w e^2z + w = e^2z - 1

e^2z(w - 1) = -1 - w

e^2z = (1 + w)/(1-w)

z = (1/2) log((1 + w)/(1-w))

arctanh(w) = (1/2) log((1 + w)/(1-w))

∫cos(x) dx /(4-sin^2(x)) = (1/2) arctanh(sin(x))/2)

= (1/4) log((1 + sin(x)/2)/(1-sin(x)/2)

= (1/4) log((2 + sin(x))/(2 - sin(x)) + C

Evaluate from 0 to 1

(1/4) log((2 + sin(1))/(2 - sin(1)) - (1/2) log((2 + sin(0))/(2 - sin(0))

= .22429 - 0 = .22429

= .448585

Source(s): if you've got an arctan button on your calculator, you can change limits then substitute