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

$\displaystyle \int u\,dv = uv - \int v\, du$


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 (a2u2), (a2 + u2), and (u2a2) 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...

  • (a2u2), try u = a sin θ
  • (a2 + u2), try u = a tan θ
  • (u2a2), 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...

$\dfrac{2x^2 + 4x - 5}{5x^3 + 6x^2 -2x -1}$


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...

$\dfrac{3x^2 - 2x + 1}{2x^2 + 6}\,\,$ and $\,\,\dfrac{4x^2 - 2x + 3}{3x + 2}$


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

$\dfrac{12x^2 - 13x - 9}{4x - 7} = 3x + 2 + \dfrac{5}{4x - 7}$


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.