Integration by parts is a powerful technique for tackling tricky integrals. It's especially useful when you've got a product of functions where one is easier to integrate and the other is easier to differentiate.

This method stems from the product rule for derivatives, flipped on its head. By breaking down complex integrals into simpler parts, it opens up new possibilities for solving previously impossible problems.

Integration by Parts

Scenarios for integration by parts

• Integrand is a product of two functions where one is easier to integrate (usually a polynomial or logarithm) and the other is easier to differentiate (usually an exponential, trigonometric, or inverse trigonometric function)
  • $\int x \sin(x) dx$ polynomial multiplied by trigonometric function
  • $\int x e^x dx$ polynomial multiplied by exponential function
  • $\int \ln(x) dx$ logarithmic function alone
• Integrand contains a logarithmic function multiplied by another function
  • $\int x \ln(x) dx$ polynomial multiplied by logarithmic function
• Integrand contains an inverse trigonometric function multiplied by another function
  • $\int x \arctan(x) dx$ polynomial multiplied by inverse trigonometric function
• Other integration techniques (u-substitution, partial fractions) are not applicable or effective for the given integral

Application of integration-by-parts formula

• Integration-by-parts formula $\int u dv = uv - \int v du$ derived from the product rule for differentiation
• Apply the formula by following these steps:
  1. Identify the integrand as a product of two functions $u$ and $dv$
  2. Differentiate $u$ to find $du$
  3. Integrate $dv$ to find $v$ (this step involves finding an antiderivative)
  4. Substitute expressions for $u$, $du$, $v$, and $dv$ into the integration-by-parts formula
  5. Simplify the resulting expression and evaluate the new integral
• Some cases require multiple applications of integration by parts to solve the integral completely (repeated integration by parts)
  • $\int x^2 e^x dx$ requires two applications of integration by parts

Definite integrals using integration by parts

• Evaluate a definite integral using integration by parts with these steps:
  1. Apply the integration-by-parts formula to the indefinite integral
  2. Evaluate the resulting expression at the given limits of integration
  3. Subtract the evaluated expression at the lower limit from the evaluated expression at the upper limit
• When setting up limits of integration use the original limits given in the problem
• Substitute limits into the entire resulting expression not just the remaining integral
• If the resulting expression after applying integration by parts contains another definite integral evaluate that integral separately using the same limits of integration
• Simplify the final answer and express it in terms of the given limits of integration

Related Concepts and Applications

• Integration by parts is closely related to the fundamental theorem of calculus, which connects differentiation and integration
• This technique is often used in solving differential equations, particularly those involving products of functions
• In some cases, integration by parts can be combined with other methods like u-substitution for more complex integrals