What is partialfraction (t-1)/(t^2+2t+1) ?

The solution to partialfraction (t-1)/(t^2+2t+1) is 1/(t+1)-2/((t+1)^2)

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partialfraction (t-1)/(t^2+2t+1)

partial fractions

\frac{t - 1}{t ^{2} + 2 t + 1}

\frac{1}{t + 1} - \frac{2}{( t + 1 ) ^{2}}

Solution steps

\frac{t - 1}{t ^{2} + 2 t + 1}

Factor

t^{2} + 2 t + 1 : (t + 1)^{2}

= \frac{t - 1}{( t + 1 ) ^{2}}

Create the partial fraction template using the denominator

(t + 1)^{2}

\frac{t - 1}{( t + 1 ) ^{2}} = \frac{a _{0}}{t + 1} + \frac{a _{1}}{( t + 1 ) ^{2}}

Multiply equation by the denominator

\frac{( t - 1 ) ( t + 1 ) ^{2}}{( t + 1 ) ^{2}} = \frac{a _{0} ( t + 1 ) ^{2}}{t + 1} + \frac{a _{1} ( t + 1 ) ^{2}}{( t + 1 ) ^{2}}

Simplify

t - 1 = a_{0} (t + 1) + a_{1}

Solve the unknown parameters by plugging the real roots of the denominator:

- 1

For the denominator root

- 1 : a_{1} = - 2

a_{1} = - 2

Plug in the solutions to the known parameters

t - 1 = a_{0} (t + 1) + (- 2)

Expand

t - 1 = a_{0} t + a_{0} - 2

Group elements according to powers of

t

1 \cdot t - 1 = a_{0} t + (a_{0} - 2)

Solve

a_{0} - 2 = - 1

for

a_{0}

a_{0} = 1

Plug the solutions to the partial fraction parameters to obtain the final result

\frac{1}{t + 1} + \frac{( - 2 )}{( t + 1 ) ^{2}}

Simplify

\frac{1}{t + 1} - \frac{2}{( t + 1 ) ^{2}}

\int x sin (x + y) d x

(1 + x^{3})^{2020}

\frac{\partial}{\partial x} (arctan (x y))

\frac{\partial}{\partial x} (5 z^{3} cot (x z))

f (x) = 15 3 x

What is partialfraction (t-1)/(t^2+2t+1) ?
The solution to partialfraction (t-1)/(t^2+2t+1) is 1/(t+1)-2/((t+1)^2)