Ackbach said:The manipulations you've done there are all fine. You just need to obey the rules of algebra. Algebra doesn't go away when you start studying calculus - indeed, calculus packs down your algebra!
U substitution is a technique used in calculus to simplify integrals by replacing a complex expression with a single variable, u. This allows for easier integration and solving of the integral.
U substitution is typically used when the integrand (the expression being integrated) contains a function and its derivative, or when the integrand can be rewritten in the form of a composite function.
When choosing a u substitution, look for a part of the integrand that resembles the derivative of another part of the integrand. This will typically involve a chain rule or power rule relationship. You can also try to rewrite the integrand in the form of a composite function.
The steps for solving a u substitution problem are as follows:
1. Identify the appropriate u substitution.
2. Substitute u for the appropriate expression in the integrand.
3. Rewrite the remaining integral in terms of u.
4. Integrate the new integral with respect to u.
5. Substitute back in the original expression for u.
Some common mistakes to avoid when using u substitution include:
- Forgetting to change the limits of integration when substituting u.
- Incorrectly choosing the u substitution, resulting in a more complex integral.
- Forgetting to substitute back in the original expression for u after integrating.
- Not simplifying the integrand before substituting in u.
- Using u substitution when it is not necessary or appropriate for the given integral.