91Ó°ÊÓ

Integration by substitution is a powerful technique used to simplify complicated integrals by changing variables. It is particularly useful when dealing with composite functions, such as when there is an expression inside a function that complicates the integration. The concept is analogous to the chain rule in differentiation. To apply integration by substitution, you typically:

• Identify a part of the integrand that can be easily switched to a new variable (usually seen as a function followed by its derivative).
• Choose a suitable substitution, often denoted by setting the troublesome part equal to a new variable, say \( u \).
• Differentiate your substitution to express everything in terms of \( du \) and adjust your integral accordingly.
• Solve the new integral in simpler form, then return to the original variables by back-substituting.

In our original exercise, choosing \( u = 2x^3 + 1 \) simplified both the expression under the square root and the remaining part of the integrand, allowing for straightforward substitution.

The power rule for integration is a fundamental tool that allows us to integrate functions of the form \( x^n \). This rule states that for any real number \( n \) other than \( -1 \), the integral of \( x^n \) with respect to \( x \) is:\[ \int x^n \, dx = \frac{x^{n+1}}{n+1} + C, \]where \( C \) is the constant of integration. If \( n = -1 \), the integral simplifies to the natural logarithm \( \ln{|x|} + C \).
In our problem, once the substitution has been made, the expression transforms into \( u^{-1/2} \). Here, the power rule applies by recognizing \( n = -1/2 \). Solving for this using the power rule provides the integral value, transitioning smoothly back from \( u \) to \( x \) through back-substitution.

There are various integration techniques, but knowing which to apply comes with practice and understanding of the integrand form:

• Integration by Parts: Useful when the integrand is a product of two functions where one can be easily differentiated and the other integrated.
• Trigonometric Integration: Involves identities and substitutions when dealing with trigonometric functions.
• Partial Fraction Decomposition: Beneficial for rational functions where the numerator degree is less than the denominator.
• Integration by Substitution: As detailed earlier, great for reversing the effects of the chain rule.

For the given problem, integration by substitution was the most effective approach because it simplified the integrand from a complicated rational expression into a more basic power function of \( u \). This showcases the importance of selecting the right technique based on the form of the integrand, cruising through to a solution quickly and effectively.