Indefinite Integrals Summary

f(x) is called the integrand, F(x) is called the particular integral and C the constant of integration.

Method of substitution:

If the integrand f(x) of the integral is not in an integral form the variable of integration x is changed to a suitable variable z by substitution and on differentiation and simplification, the new integral is found integrable.

Standard integrals

By method of completing squares ax² + bx + c is expressed as A² - X² or X² - A² or A² + x² and the integral reduces to

which can be evaluated using the standard integrals.

Method:

Step 4: The second integral can be evaluated by method completing squares.

Standard integrals

By method of completing squares ax² + bx + c is expressed as A²+ X² or X² - A² and the integral reduces to

which can be evaluated using the standard integrals.

Method:

Step 3: The second integral can be evaluated by method of completing squares.

Method

Step 3: Resulting integral is evaluated by method of completing squares.

Method

Step 2: Determine L and M

Step 4: Integral = Lx + M log (a cos x + bsinx)

Integration by partial fractions:

Step 4: Integrate each part on the right hand side to obtain the required integrals.

Integration by parts

In words: Integral of the product of two functions

• If the integrand is the product of two functions of different types then their order is determined by the word ILATE where

I = Inverse trigonometric

L = Logarithmic

A = Algebraic, T = Trigonometric, E = Exponential

In the integrand, the first function is the function is the function which comes first in the word ILATE.

However, there is no rigid rule in this that you have to select the first function in this order.

• Standard integrals

Step 1: By method of completing squares

Step 2: Use standard and integrals and evaluate

Step 1: put px + q = L (2ax + b) + M and determine L and M.

Step 3: Second integral on the right hand side can be evaluated by method of completing squares.