robertjford80 said:
U substitution and integration by parts are two commonly used integration techniques in calculus. U substitution involves replacing a variable in an integral with a new variable to simplify the integral. Integration by parts is used to integrate the product of two functions by using the product rule from differentiation.
U substitution is typically used when the integrand (the expression inside the integral) contains a function and its derivative, or when it contains a polynomial function and a trigonometric function. It can also be used when the integrand contains a complicated expression that can be simplified by substituting a new variable.
The key to choosing the correct function for U substitution is to look for a function and its derivative in the integrand. The function that is being multiplied by its derivative is usually a good choice for U. If the integrand contains a polynomial and a trigonometric function, the polynomial is typically chosen as U.
To use integration by parts, you must first identify which function will be u (the one that you will differentiate) and which function will be dv (the one that you will integrate). Then, use the formula ∫u dv = uv - ∫v du to solve the integral. You may need to use integration by parts multiple times or combine it with other integration techniques to fully solve the integral.
Integration by parts is typically used when the integrand is a product of two functions, or when it contains a function that cannot be easily integrated. It is also useful when the integrand contains a polynomial function and a trigonometric function, as this can often be simplified using integration by parts.