Home > Calculus > Integral by Partial Fractions

How do you integrate #1/(y(1-y))#?

Answer 1

Emilia Aguilar

Have a look:

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 2

Emily Ammerman

#int1/(y(1-y))dy=-int1/(y^2-y+(-1/2)^2-(-1/2)^2)dy#

#=>-int1/((y-1/2)^2-(1/2)^2)dy=-1/(1/2)^2int1/(((y-1/2)/(1/2))^2-1)dy=-4int1/((2y-1)^2-1)dy#

#=>1/2*-4artanh(2y-1) + C = -ln|(1-2y+1)/(1+2y-1)| + C=ln|(y)/(1-y)| + C#

By signing up, you agree to our Terms of Service and Privacy Policy

Answer 3

Emma Aldridge

To integrate ( \frac{1}{y(1-y)} ), we can use partial fraction decomposition.

We decompose the fraction into the form ( \frac{A}{y} + \frac{B}{1-y} ).

Then, we find ( A ) and ( B ) by equating the original expression to the decomposed expression.

After finding ( A ) and ( B ), we integrate each term separately, which results in ( A\ln|y| - B\ln|1-y| + C ), where ( C ) is the constant of integration.

By signing up, you agree to our Terms of Service and Privacy Policy

Answer from HIX Tutor

When evaluating a one-sided limit, you need to be careful when a quantity is approaching zero since its sign is different depending on which way it is approaching zero from. Let us look at some examples.