Home » Partial Fractions » Page 2

# Evaluate the integral of 1 / (x2 – 1)2

Compute the following integral.

First, we write

Then we use partial fractions, writing

This gives us the equation

Evaluating at and we obtain the values for and ,

Using these values of and and evaluating at and we have

Solving this system for and we get

Thus,

# Evaluate the integral of (x-3) / (x3 + 3x2 + 2x)

Compute the following integral.

The denominator of the integrand factors as

Therefore, we can use partial fractions as follows,

This gives us the equation

Evaluating at , , and we obtain

Therefore, we have

# Evaluate the integral of 1 / ((x2 – 4x + 4)(x2 – 4x + 5))

Compute the following integral.

In the denominator we have

Then we use partial fractions,

This gives us the equation

We evaluate at to obtain a value for ,

Then using this value of and evaluating at , and to obtain

Solving this system of equations we obtain

Therefore, we have

# Evaluate the integral of x2 / (x2 + x – 6)

Compute the following integral.

First, we have

Therefore,

We use partial fractions to evaluate the integral on the right. To that end, we write

This gives us the equation

Evaluating at and we then have

Therefore,

# Evaluate the integral of 1 / (x3 – x)

Compute the following integral.

Since factors as we have the following,

This gives us the equation

Substituting the values , and we obtain

Therefore, we have

# Evaluate the following integral 1 / ((x+1)(x+2)2(x+3)3)

Compute the following integral.

First, we use partial fraction decomposition. We write

This gives us the equation

First, we substitute the values , , and which gives us

Substituting these values of , and into our equation we have

Now, we substitute the values , and to obtain the equations

Solving this system we obtain the values

Therefore, we have the following,

# Evaluate the integral of 1 / (x (x2 + 1)2)

Compute the following integral.

To evaluate this we use partial fractions. First, we write

This gives us the equation

First, we evaluate at to find that . Using this value of we obtain the equation

Equating like powers of this gives us the system of equations:

We solve this system to obtain,

Therefore,

# Evaluate the following integral (x+2) / (x2 + x)

Compute the following integral.

We have

To evaluate the second integral on the right we use partial fractions. Write,

This gives us the equations

Letting and we have

Therefore, we have

# Evaluate the integral of x4 / (x4 +5x2 + 4)

Compute the following integral.

First, we simplify the integrand and then use partial fractions,

To evaluate the second integral we use partial fractions. We have

Therefore, we write

Then we have the equation

Equating like powers of we then have four equations and four unknowns:

Solving this system we find , and . Therefore, we have

# Evaluate the integral of (8x3 + 7) / ((x+1)(2x+1)3)

Compute the following integral.

Since the denominator is already factored into linear terms we can proceed directly with the partial fraction decomposition. We write,

This gives us the equation

First, we can find the values of and by evaluating at and , respectively. This gives us

Then using these values of and we evaluate at and (these are just convenient values, there isn’t anything special about them) to obtain the two equations

Solving these two equations we obtain and . Therefore, we have the following partial fraction decomposition:

We can now evaluate the integral,