91Ó°ÊÓ

The substitution method is an important integration technique in calculus, especially useful when dealing with complex integrands. The goal is to simplify the integral by introducing a new variable.
In this problem, we used the substitution method by setting \( u = \sqrt[4] x +1 \). This transformation helps break the integral into more manageable parts. By differentiating and solving for \( dx \), we can rewrite the integral in terms of the new variable \( u \).
This new substitution changes the integral's form and makes computation easier. Without it, solving directly would be more complicated, often obscuring the method’s beauty and utility.

Integration techniques include a variety of methods used to evaluate integrals. Methods like substitution and integration by parts are key tools.
For this problem, after substitution, we expanded and simplified the expression. By breaking it into simpler terms, we applied the power rule of integration to each term separately:

• <\( u^{3/2} \ int -> \ int \frac{2}{5}u^{5/2} \)
• <\( u^{1/2} \ int -> \ int u^{3/2} \)
• and so on.
  Such techniques often involve algebraic manipulation. Results are combined to arrive at the answer and then the original variables are reinstated to get the final expression.

Calculus is the branch of mathematics that deals with continuous change. Integrals and derivatives are its main tools.
In the context of this problem, integration or finding the antiderivative was key. Integrals can tell us information like area under curves, which translates to accumulated values in various fields.
Through steps like substitution, simplification, and applying integration rules, calculus provides means to solve complex problems efficiently. Understanding these foundational steps aids immensely in tackling higher-level mathematics and applied problems. Applying such methods repeatedly in various problems solidifies comprehension and broadens problem-solving capabilities.