Question

How do I know when to use integration by partial fraction?

Answer 1

Hint:
Partial fractions is a fraction that can possibly split many fractions. The method of partial fractions basically allows us to split the right hand side of the equation into the left hand side. Partial fractions can be used to turn the functions that cannot be integrated into simple fractions easily.

Complete step by step solution:
Partial fractions can be used in Mathematics to turn functions that cannot be integrated into simple fractions easily. We can basically use the partial fractions if the degree of the numerator is strictly less than the degree of the denominator.
The concept of Partial fractions works only in the case of proper rational expressions. A proper rational expression is one where the degree of the numerator is less than the denominator.
If the expression is in the form of partial fraction or improper fraction, then we need to use polynomial division method first.
And the process of taking a rational expression and decomposing it into simpler rational expressions that we can add or subtract to get the original rational expression is called partial fraction decomposition. Many integrals involving rational expressions can be done if we first do partial fractions on the integrand.
Determining whether or not to use partial fractions is easy. If our substitution can't be used on the rational function, we can use partial fractions, but we have to make sure the numerator is of a smaller degree than the denominator.

$\therefore $ We can basically use the partial fractions if the degree of the numerator is strictly less than the degree of the denominator. If the expression is in the form of partial fraction or improper fraction, then we need to use polynomial division method first.

Note:
We can solve the partial fraction using following steps:
1) Start with Proper Rational expressions.
2) We need to factor the denominator into linear factors.
3) Now we need to write out a partial fraction for each factor.
4) Next multiply the whole equation by the denominator.
5) For solving the coefficients we need to substitute zeros of the denominator.