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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

    Improper integrals

    \(\displaystyle \int_1^\infty 4x^{-4}\,dx\)
    \(\dfrac{4}{3}\)
    10550Improper Integrals 5 Examples Calculus 2 BC
    ProfRobBob
    \(\displaystyle \int_1^\infty 2x^{-1 / 5}\,dx\)
    No solution.
    10550Improper Integrals 5 Examples Calculus 2 BC
    ProfRobBob
    \(\displaystyle \int_{-\infty}^\infty \dfrac{4}{36 + x^2}\,dx\)
    \(\dfrac{2\pi}{3}\)
    10550Improper Integrals 5 Examples Calculus 2 BC
    ProfRobBob
    \(\displaystyle \int_0^{e^2} \ln x^3\,dx\)
    \(3e^2\)
    10550Improper Integrals 5 Examples Calculus 2 BC
    ProfRobBob
    \(\displaystyle \int_3^{9} \dfrac{1}{\sqrt{x^2 - 9}}\,dx\)
    \(\ln(3 + 2\sqrt{2})\)
    10550Improper Integrals 5 Examples Calculus 2 BC
    ProfRobBob

    Conclude by giving your students these challenges:

    • Gambling at Monte Carlo by NRICH
    • Rati-o by NRICH
    • Percent Shaded

    Place all the digit cards to make all four equations simultaneously true.

    What Two ...? by NRICH: Let students know they'll need a calculator to solve this problem.

    YouTube videos

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