Substitution is a technique used to evaluate integrals by replacing the variable of integration with a new variable. This new variable is chosen in such a way that it simplifies the integral and makes it easier to solve.
Substitution should be used when the integral contains a complicated or nested function. It also works well when the integral contains a product of functions or when the power of a function is not an integer.
The substitution variable should be chosen in such a way that it simplifies the integral. A common approach is to choose a variable that is inside a function, such as the argument of a trigonometric function or inside a radical.
The general process for evaluating an integral using substitution is as follows:1. Identify the most suitable substitution variable.2. Rewrite the integral in terms of the new variable.3. Calculate the differential of the new variable.4. Substitute the new variable and its differential into the integral.5. Solve the resulting integral.
Yes, there are a few tips that can make using substitution easier:- Choose the substitution variable carefully, it can greatly impact the simplicity of the integral.- Remember to substitute back in the original variable at the end.- Keep track of the differential when substituting.- Check your answer by differentiating it to see if it matches the original integrand.