# 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:

1. Choose u and dv:

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

• Calculate du and v by differentiating u and integrating dv.
3. Apply the Formula:

• Substitute the values of u, du, v, and dv into the integration by parts formula.
4. 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:

1. Choose u and dv:

• Let $u=x$ (easily differentiable)
• $dv={e}^{x}dx$ (integral of ${e}^{x}$ is ${e}^{x}$)
2. Differentiate and Integrate:

• Calculate $du=dx$ and $v={e}^{x}$
3. Apply the Formula:

• Substitute into the integration by parts formula:

$\int x\cdot {e}^{x}dx=x\cdot {e}^{x}-\int {e}^{x}\cdot dx$

4. 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.