Integration using Partial Fractions

Example 1:

Evaluate.

Note: We cannot simply the denominator since the denominator expands to , which doesn’t differentiate to the form of the numerator. It is important that we always check this.

Step 1: Write in partial fractions.

We rewrite the RHS to match the same form as the LHS by cross-multiplying.

Equating the numerators, we solve for and by substituting suitable values for that makes one of the brackets equal to .

So, we have

Therefore, we are evaluating

Step 2: Make guesses as to what may differentiate to give each term in our integral.

• For , we will guess . The is a multiplier so we can ignore it in our guess.

• For , we will guess . Like before, we can ignore the in our guess because that’s a multiplier.

Step 3: Differentiate our guesses using the chain rule.

• For , the angle is the argument (what we’re taking of). In this case, the angle is
, which differentiates to . Then, when we differentiate the function of , the angle is reciprocated.

When is differentiated, the angle is reciprocated



Differentiated angle

• For , the angle is , which differentiates to . Then, when we differentiate the function of , the angle is reciprocated.

When is differentiated, the angle is reciprocated



Differentiated angle

Step 4: Adjust the constants.

• Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .

• Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .

Example 2:

a) Express in partial fractions.

Here, we have a fraction that has a repeated factor in the denominator, namely , so we express the partial fractions as

Step 1: Write the RHS as a single fraction. We want the denominator to be . We don’t cross-multiply in this case. Instead, we multiply the numerator and denominator of by , by and by .

We get:

Step 2: Equate the numerators and solve for , , and by substituting suitable values for that make one of the brackets equal to .

We will run out of options on what to substitute to eliminate one of the unknowns, so we choose the next easiest option which is .

Substitute our values of and ,

Now rearrange for ,

So, we have

b) Hence, find

We are evaluating

To note, the term is a power function and can be rewritten as .

Step 1: Make guesses as to what may differentiate to give each term in our integral.

• For , we will guess . The is a multiplier so we can ignore it in our guess.

• For , we will guess . Similarly, is a multiplier so we can ignore it in our guess.

• For , we will guess . With power functions, the power goes up by . The is a multiplier so we can ignore it in our guess.

Step 2: Differentiate the guesses using the chain rule.

• For , the angle is the argument (what we’re taking of). In this case, the angle is
, which differentiates to . Then, when we differentiate the function of , the angle is reciprocated.

When is differentiated, the angle is reciprocated



Differentiated angle

• For , the angle is , which differentiates to . Then, when we differentiate the function of , the angle is reciprocated.

When is differentiated, the angle is reciprocated



Differentiated angle

• For , the angle is what’s inside the bracket, which is in this case. The angle differentiates to give . Then, for the power function, we bring down the power, knock off the power, then keep the angle the same.

Angle stays the same



Differentiated angleDifferentiated power function

Step 3: Adjust the constants.

• Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which here will be .

• Our guess differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which here will be .

• Our differentiates to give , but we wanted . So, we multiply both sides by the constant we want over the constant we have, which will be .

Therefore, we are left with a final answer of,

Then if you want to rewrite without negative powers (not necessary) we will have,