## Integration by Parts

When $u$ and $v$ are differentiable functions of $x$, $d(uv) = u \, dv + v \, du\,$ or $\,u \, dv = d(uv) - v \, du$. When this is integrated we have

The expression to be integrated must be separated into two parts, one part being $u$ and the other part, together with $dx$, being $dv$. The factor corresponding to $dv$ must obviously contain the differential of the variable of integration.

## Integration by Substitution

There are two types of substitution: algebraic substitution and trigonometric substitution.

### Algebraic Substitution

In algebraic substitution we replace the variable of integration by a function of a new variable. A change in the variable on integration often reduces an integrand to an easier integrable form.

### Trigonometric Substitution

Trigonometric substitution is employed to integrate expressions involving functions of (*a*^{2} − *u*^{2}), (*a*^{2} + *u*^{2}), and (*u*^{2} − *a*^{2}) where "*a*" is a constant and "*u*" is any algebraic function. Substitutions convert the respective functions to expressions in terms of trigonometric functions. The substitution is more useful but not limited to functions involving radicals.

**Use the following suggestions:**

When the integrand involves...

- (
*a*^{2}−*u*^{2}), try*u*=*a*sin θ - (
*a*^{2}+*u*^{2}), try*u*=*a*tan θ - (
*u*^{2}−*a*^{2}), try*u*=*a*sec θ

The substitution may be represented geometrically by constructing a right triangle.

## Integration of Rational Fractions

**Partial Fraction**

Functions of *x* that can be expressed in the form *P*(*x*)/*Q*(*x*), where both *P*(*x*) and *Q*(*x*) are polynomials of *x*, is known as **rational fraction**. A rational fraction is known to be a **proper fraction** if the degree of *P*(*x*) is less than the degree of *Q*(*x*). Example of proper fraction is...

A rational fraction is said to be an improper fraction if the degree of *P*(*x*) is greater than or equal to the degree of *Q*(*x*). Examples are...

Improper fraction may be expressed as the sum of a polynomial and a proper fraction. For example:

Proper fraction such as $\dfrac{x - 4}{2x^2 - 4x}$ can be expressed as the sum of *partial fraction*, provided that the denominator will *factorized*.

Integration of any rational fraction depends essentially on the integration of a proper fraction by expressing it into a sum of partial fractions. There are four cases that may arise in dealing with integrand involving proper fraction.