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    • ▾Calculus 2
      • ▾Integration techniques
        • •Intro to calc 2
        • ▸Basic indefinite integrals
          • •Reverse power rule for indefinite integrals
          • •Sum, difference, and reverse power rules for indefinite integrals
        • •Integration by parts
        • ▸Integrals involving trig functions
          • •Lesson
          • •Practice
        • •Trig substitutions
        • •Integration by partial fractions
        • •Integral recurrence relations
        • •Integrating quadratics by completing the square
        • •Integration by long division
        • •Improper integrals
        • •Comparison tests for convergence
        • ▸Approximating definite integrals
          • •Riemann sums
          • ▸Simpson's rule
            • •Practice: Simpson's rule
          • ▸Trapezoidal rule
            • •Lesson
            • •Practice
          • •Midpoint rule
      • ▸Applications of integrals
        • ▸Arc length (calculus 2)
          • •Using the formula
        • ▸Center of mass (calculus 2)
          • •Center of mass for one and two-dimensional systems
          • •Center of mass for planar lamina
          • •Pappus's theorem
        • •Hydrostatic force
        • •Mode of a continuous random variable from a probability density function
        • •Surface area of solids of revolution
        • •Gabriel's Horn
        • •Applications of trigonometric integrals
        • •Integrating exponential functions
        • •Logarithmic functions
      • ▸Parametric equations and polar coordinates
        • •Cycloid area and length
        • •Eliminating the parameter
        • •Testing polar equations for symmetry
        • •Deriving the equations of a cycloid, epicycloid, and hypocycloid
      • ▸Series and sequences
        • •Maclaurin series
        • •Finding the sum of a telescoping series
        • ▸Convergent and divergent geometric series
          • •Lesson
          • •Practice
        • •Alternating harmonic series
        • •Alternating series test
        • •Partial sums
        • •Ratio test
        • ▸Root test
          • •Lesson
          • •Practice
        • •Absolute convergence, conditional convergence, and divergence
        • •Riemann's rearrangement theorem
        • •Taylor series
      • ▸Vectors
        • •Intro to vectors
        • •Length of a vector
        • •Cross product
        • •Intro to the dot product
      • •Competition problems (calculus 2)
     › Calculus 2 › Integration techniques

    Integration by parts

    First, students will learn how the integration by parts formula is derived from the product rule.

    Integration by parts intro by Khan Academy

    Next, students will learn how to do integration by parts for both definite and indefinite integrals. They will also learn the tabular method for integration by parts. Here's a video covering definite integration by parts. Here's a video covering the tabular method.

    Conclude by giving your students these challenges:

    • Children at Large by NRICH
    • Maths Is Everywhere! by NRICH
    • 2007 AMC 8, Problem 21
    • Arithmetic Logs

    Mentally determine the length of \(\overline{CD}\,{:}\)

    Here's the solution.

    Practice problems and additional lessons