steve092 said:Thanks for the help. I got:
(4/7)(x+2)^(7/4) - (8/3)(x+2)^(3/4) +C.
Substitutions are used in integrals to simplify the expression and make it easier to solve. In this particular expression, the substitution helps to eliminate the fractional exponent, making it easier to integrate.
The choice of substitution depends on the structure of the integral. In this case, we need to choose a substitution that will eliminate the fractional exponent. A common approach is to let u = (x+2)^(1/4), then we can replace x with u^4 in the original expression.
Yes, it is possible to use multiple substitutions in an integral. However, it is important to ensure that the substitutions are compatible with each other and do not introduce any new complexities to the integral.
Substitutions help to simplify integrals and make them easier to solve. They can also help to reveal patterns and make connections between different types of integrals. Additionally, substitutions can also be used to transform integrals into a standard form, making it easier to apply known integration techniques.
Substitutions may not always work for every type of integral. In some cases, they may introduce complexities or result in an unsolvable integral. It is important to carefully consider the structure of the integral before choosing a substitution. Additionally, some integrals may require multiple substitutions or other techniques in combination with substitutions to be solved effectively.