Integration by Parts
Integration by Parts Method for Indefinite Integrals:
 Principle: Used for integrating the product of two functions.
 Formula: ∫ u dv = uv  ∫ v du, where u and v are functions of x.
 Derivation: Derived from the product rule of differentiation (d(uv)/dx = u dv/dx + v du/dx).
Steps for Integration by Parts:

Choose u and dv:
 u: Select a function that can be easily differentiated.
 dv: Choose a function whose integral can be easily computed.

Differentiate and Integrate:
 Calculate du and v by differentiating u and integrating dv.

Apply the Formula:
 Substitute the values of u, du, v, and dv into the integration by parts formula.

Solve for the Integral:
 Evaluate the resulting integral, which may involve repeating the integration by parts method or simplification.
Example:
Let's solve the indefinite integral ∫ x * ${e}^{x}$ dx using the integration by parts method:
Steps:

Choose u and dv:
 Let $u=x$ (easily differentiable)
 $dv={e}^{x}dx$ (integral of ${e}^{x}$ is ${e}^{x}$)

Differentiate and Integrate:
 Calculate $du=dx$ and $v={e}^{x}$

Apply the Formula:
 Substitute into the integration by parts formula:
$\int x\cdot {e}^{x}dx=x\cdot {e}^{x}\int {e}^{x}\cdot dx$

Solve for the Integral:
 The integral of ${e}^{x}$ is ${e}^{x}$, so the solution is:
$x\cdot {e}^{x}{e}^{x}+C$
Tips for Integration by Parts:
 Choose u and dv Wisely: Select u and dv in such a way that du or v can be easily computed.
 Usefulness in Repeated Integration: Sometimes, repeated application of integration by parts might be necessary to solve the integral completely.
 Identify Patterns: Look for patterns that might simplify the integral or make it easier to solve.