10.2 Integration by Parts

Integration by parts is a powerful technique for solving complex integrals. It's especially useful for products of functions that are tricky to integrate directly, like polynomials with trig functions or exponentials.

The LIATE rule helps you choose which part of the integrand to differentiate and which to integrate. This method, along with the tabular approach, makes solving these integrals more systematic and less prone to errors.

Integration Techniques

Product Rule and LIATE Rule

Top images from around the web for Product Rule and LIATE Rule

• 

  Integration and The Fundamental Theorem of Calculus - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Unit 2: Rules for integration – National Curriculum (Vocational) Mathematics Level 4 View original

  Is this image relevant?

• 

  hyperbolic functions - Integration by parts - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Integration and The Fundamental Theorem of Calculus - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Unit 2: Rules for integration – National Curriculum (Vocational) Mathematics Level 4 View original

  Is this image relevant?

1 of 3

Top images from around the web for Product Rule and LIATE Rule

• 

  Integration and The Fundamental Theorem of Calculus - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Unit 2: Rules for integration – National Curriculum (Vocational) Mathematics Level 4 View original

  Is this image relevant?

• 

  hyperbolic functions - Integration by parts - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Integration and The Fundamental Theorem of Calculus - Mathematics Stack Exchange View original

  Is this image relevant?

• 

  Unit 2: Rules for integration – National Curriculum (Vocational) Mathematics Level 4 View original

  Is this image relevant?

1 of 3

• The product rule for integration by parts states $\int u\,dv = uv - \int v\,du$
• Requires choosing $u$ and $dv$ from the integrand $f(x)g(x)\,[dx](https://www.fiveableKeyTerm:dx)$
• LIATE is a mnemonic for choosing $u$:
  • L ogarithmic functions
  • I nverse trigonometric functions
  • A lgebraic functions (polynomials, rational functions)
  • T rigonometric functions (sine, cosine, tangent, etc.)
  • E xponential functions
• Choose the higher priority function as $u$ according to LIATE and the other as $dv$
• After choosing $u$ and $dv$, compute $du$ and $v$ before substituting into the product rule formula
• Example: $\int x\cos x\,dx$ with $u=x$ and $dv=\cos x\,dx$ since algebraic functions have higher priority than trigonometric ones

Tabular Method and Recursive Integration

• The tabular method organizes the integration by parts steps into a table
• Helps to avoid mistakes and clearly shows the pattern for integrals requiring multiple integration by parts steps
• Set up the table with $u$ and $du$ in one column and $v$ and $dv$ in the other
• Work down the column, differentiating $u$ to get $du$ and integrating $dv$ to get $v$
• Compute the products $uv$ along the diagonals and add/subtract with alternating signs
• Recursive integration refers to repeating the integration by parts process on the remaining integral term
• Commonly needed for integrals involving products of polynomials and trigonometric or exponential functions
• Example: $\int e^x\cos x\,dx$ requires recursive integration by parts, first with $u=\cos x$ and $dv=e^x\,dx$, then with the resulting integral of $\int e^x\sin x\,dx$

Functions

Exponential and Logarithmic Functions

• Integration by parts is often used for integrals involving exponential and logarithmic functions
• For exponential functions of the form $e^{ax}$, choosing $u=e^{ax}$ and $dv=dx$ frequently works well
  • Leads to $du=ae^{ax}\,dx$ and $v=x$, simplifying the resulting integral
• For logarithmic functions of the form $\ln(x)$, choosing $u=\ln(x)$ and $dv=dx$ is a common strategy
  • Gives $du=\frac{1}{x}\,dx$ and $v=x$, allowing for a simplification
• Example: $\int e^{3x}x^2\,dx$ can be solved by taking $u=x^2$ and $dv=e^{3x}\,dx$

Trigonometric Functions

• Integration by parts is frequently applied to integrals involving products of trigonometric and algebraic functions
• For products like $x\sin(x)$ or $x^2\cos(x)$, choosing the algebraic component as $u$ usually works best
  • Leads to polynomial expressions for $du$ and trigonometric expressions for $v$
• Integrals with products of trigonometric functions often require recursive integration by parts
  • Successive steps alternate between sine and cosine as the $u$ term
• Useful trigonometric integral formulas to remember:
  • $\int \sin(x)\,dx = -\cos(x) + C$
  • $\int \cos(x)\,dx = \sin(x) + C$
• Example: $\int x\sin(3x)\,dx$ can be solved using $u=x$ and $dv=\sin(3x)\,dx$, leading to $du=dx$ and $v=-\frac{1}{3}\cos(3x)$