An integral is a mathematical concept that represents the calculation of the area under a curve on a graph. It is used to find the total value of a continuously changing quantity, such as distance or velocity.
Substitution methods are used to simplify complicated integrals by substituting a variable with a simpler expression. This makes the integral easier to solve using basic integration rules.
If the integral contains a function within a function, or if it can be rewritten in a simpler form, then a substitution method is likely needed. Another indication is if the integral involves trigonometric functions, logarithms, or exponential functions.
Some common substitution methods include u-substitution, trigonometric substitution, and hyperbolic substitution. These methods involve substituting a variable or trigonometric function with a simpler expression to make the integral easier to solve.
You can check your substitution method by differentiating your final answer and seeing if it simplifies back to the original integrand. If it does, then your substitution method was correct. Additionally, you can check your answer using a graphing calculator or by using a definite integral to find the area under the curve.