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# Evaluate the integral of (x4+2x-6) / (x3 + x2 – 2x)

Compute the following integral.

First, we can factor the denominator,

Next, we want to simplify the expression. In particular, we want the numerator to be a polynomial of smaller degree than the denominator. Dividing the numerator by the denominator we obtain

Therefore, we can start evaluating the integral,

Now we need to use partial fractions to evaluate this last integral on the right. To that end we write,

Therefore, we have the equation

Substituting the values , , and we obtain the following values

So, then we can continue evaluating the integral,

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

Compute the following integral.

First, we factor the denominator,

Then we use partial fractions to decompose the integrand,

This gives us the equation

Substituting the values and

Then we use these values of and and plug in to find ,

So, now we can evaluate the integral with this partial fraction decomposition,

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

Compute the following integral

First, we need to get the partial fraction decomposition of the integrand. To that end write

Then we have the equation

Plugging in the values , , and we obtain the following

Therefore we have

# Evaluate the integral of (2x+3) / ((x-2)(x+5))

Compute the following integral.

First, we find the partial fraction decomposition of the integrand,

This implies

Plugging in the values and we obtain the equations

Therefore we have