U-Substitution (Indefinite Integrals)

Front

What is U-Substitution in integration?

Back

U-Substitution is a method used to simplify the process of integration by substituting a part of the integrand with a new variable, usually denoted as 'u', to make the integral easier to solve.

Front

When should you use U-Substitution?

Back

Use U-Substitution when the integrand is a composite function, especially when the derivative of the inner function is present in the integrand.

Front

What is the first step in U-Substitution?

Back

Identify a part of the integrand to substitute with 'u', typically the inner function of a composite function.

Front

How do you express dx in terms of du?

Back

After substituting u, differentiate u with respect to x to find du, and then express dx as du divided by the derivative of u.

Front

What is the formula for U-Substitution?

Back

If u = g(x), then ∫ f(g(x)) g'(x) dx = ∫ f(u) du.

Front

Example of U-Substitution: ∫ 6x(x² + 1)² dx. What is u?

Back

Let u = x² + 1.

Front

After substituting u in ∫ 6x(x² + 1)² dx, what is the new integral?

Back

The new integral becomes ∫ 6u² du.

Front

What is the integral of 6u² du?

Back

The integral is 2u³ + C.

Front

What is the final answer for ∫ 6x(x² + 1)² dx after back-substituting?

Back

The final answer is (x² + 1)³ + C.

Front

Example of U-Substitution: ∫ 2x cos(x²) dx. What is u?

Back

Let u = x².