Indefinite Integrals and U Substitution

Indefinite Integrals and U Substitution

• U substitution simplifies the integration of composite functions by making them manageable.
• The objective is to transform a complicated integral into an easier one by substituting one part of the integrand.

Steps to U Substitution

• Choose a variable 'u' that simplifies the integral, often the inner function of a composite function.
• Ensure the derivative of 'u' appears in the integrand to facilitate substitution.
• Rewrite the integral in terms of 'u' and 'du' (moving from f(x)dx to u(du)).
• Solve the integral before translating the result back to the original variable.

Writing the Integral in Terms of U

• Define 'u' as a suitable function (e.g. ( u = x^2 + 1 )).
• Calculate the differential ( du ) as the derivative of 'u' multiplied by ( dx ) (e.g., ( du = 2x , dx )).

Substituting dx

• Express ( dx ) in terms of ( du ): ( dx = \frac{du}{2x} ).
• Replace ( dx ) in the integral with its expression in terms of ( du ).

Quick Guide to U Substitution

• Select an appropriate 'u'.
• Determine the adjustment needed for ( dx ) through its derivative.
• Substitute into the integral and, if possible, cancel terms.
• Perform the integration of ( u ) and apply any external constants.
• Revert 'u' back to the original variable without further manipulation.

Derivative and Constant Pull-Out Rule

• The derivative involves maintaining ( dx ) as part of the expression rather than simplifying to 1.
• Use the constant pull-out rule to simplify integrals: ( k \int{\text{function}} , dx = k \cdot \int{\text{function}} , dx ).

Selecting the Appropriate U

• The derivative of 'u' must be present in the original integrand; verify by differentiation.
• The chosen 'u' should lead to a simpler integral expression.

Antiderivative and ln Rule

• The antiderivative of ( \frac{1}{u} ) is ( \ln |u| ), a crucial consideration for fractions.
• When substituting variable parts in fractions, note that ( u ) in the denominator affects the result.

New Finding Rule

• In complicated integrals, identify only the innermost operations for substitution.
• If a constant factor can be isolated during differentiation, it can remain in the setup without impacting the overall process.

Absolute Value Rule

• When dealing with fractions that involve ln, remember to use the absolute value: ( \ln |u| ).

Additional Antiderivative Notes

• The rule for the antiderivative helps differentiate terms like ( \frac{1}{u^2} ), keeping in mind the simplification leads to a negative exponent outcome.