-Substitution in Integration

Overview

• -substitution is a technique used in calculus to find antiderivatives of composite functions.
• It reverses the chain rule for differentiation.

Basic Concept

• When finding antiderivatives, reverse differentiation is performed.
• Straightforward cases involve basic functions (e.g., sin(x), ex).
• Complex integrals like cos(3x + 5)dx require -substitution.

-Substitution Process

Using -Substitution with Indefinite Integrals

1. Identify the inner function: For example, in 2xcos(x^2)dx:
   • Let u = x^2 (inner function).
2. Differentiate the inner function:
   • du = 2xdx.
3. Substitute into the integral:
   • Replace 2xcos(x^2)dx with cos(u)du.
4. Integrate and substitute back:
   • Find antiderivative of cos(u) which is sin(u) + C.
   • Substitute back to get sin(x^2) + C.

Key Takeaways

• -Substitution reverses the chain rule.
• Simplifies expressions by making the inner function the variable.

Common Mistakes

Incorrect Expression for u or du

• Choosing the wrong u results in incorrect integrals.
• For example, in (6x^2)(2x^3 + 5)^6dx, u must be 2x^3 + 5.
• Incorrect differentiation of u leads to wrong du and hence wrong results.

Ignoring Composite Functions

• Cannot simply take the antiderivative of the outer function.
• Example: cos(5x^7)dx cannot be solved as sin(5x^7) + C.

Confusing Inner Function and Its Derivative

• Ensure the derivative of the inner function matches part of the integrand.

Handling Non-Multiplicative Integrals

• Sometimes need to multiply/divide the integral by a constant for substitution.
• Example: sin(3x+5)dx becomes (1/3)sin(3x+5)(3dx).

Practice Problems

• Define u and use substitution for various integrals (e.g., incorrect options for u, multiplying by constants).

Additional Topics

• Integration by parts is suggested if -substitution fails.
• Multiple valid antiderivatives exist; differentiation confirms equivalence.

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This overview covers the essential aspects and common pitfalls of -substitution. It serves as a guide for practicing and mastering this integral solving technique.