Calculus 6.9 Integration with U-Substitution

U-substitution is a method used to simplify the integration process by substituting a part of the integrand with a new variable 'u', making the integral easier to solve.

Use u-substitution when the integrand contains a function and its derivative, allowing for a simpler integral after substitution.

Identify a part of the integrand to substitute with 'u', typically a function whose derivative is also present in the integrand.

$$du = 2x \, dx$$ .

Change the limits of integration to correspond to the new variable 'u' by substituting the original limits into the equation for 'u'.

$$\text{The integral is } \text{ln}|x| + C$$ .

It implies that if you choose $$u = ln(x)$$ , then $$du = \frac{1}{x}dx$$ , making integration easier.

$$\text{The integral is } -\frac{1}{x} + C$$ .

Changing variables simplifies the integral, making it easier to evaluate.

$$\text{The integral is } e^{u} + C$$ .