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    • ▾Calculus 1
      • ▸Limits and continuity
        • •Intro to calc 1
        • •Estimating limits from graphs
        • •Estimating limits from tables
        • •Intro to continuity
        • •Finding and classifying discontinuities
        • •Finding the value that makes the function continuous
      • ▸Derivatives: definition and basic derivative rules
        • •Proving differentiable implies continuous
        • •Derivative of a function from first principles
        • •Finding the tangent or normal line through the given point
        • •Constant rule
        • •Constant multiple rule for derivatives
        • •Sum, difference, and power rules for derivatives
        • •Product rule
        • •Proving the product rule
        • •Quotient rule
        • •Derivatives of trig functions
        • •Proving properties of even and odd functions (calculus 1)
        • •Leibniz's derivative notation
        • •Logarithmic functions
      • ▸Derivatives: composite, implicit, and inverse functions
        • •Derivative at a point from first principles
        • •Derivatives of exponential functions
        • •Derivatives of logarithmic functions
        • ▸Derivatives of hyperbolic and inverse hyperbolic functions
          • •Using the derivatives of the hyperbolic functions
          • •Proving the derivatives of the hyperbolic functions
          • •Proving the derivatives of the inverse hyperbolic functions
        • •Derivatives of inverse trig functions
        • •Derivatives of transcendental functions
        • •Derivatives of trigonometric functions under transformations
        • •Derivative using u-substitution
        • •Determining derivatives from graphs
        • •Higher order derivatives
        • •Chain rule
        • •Derivatives of inverse functions
        • •Choosing the derivative rules to use
      • ▸Applying derivatives to analyze functions
        • •Applications of trigonometric derivatives
        • •Extreme value theorem
        • •First derivative test
        • •Mean value theorem
        • ▸Monotonic intervals and functions
          • •Intro
          • •Proofs
        • •Related rates
        • •Second derivative test
        • •Concavity
        • •Factoring a cubic polynomial with a double root
        • •Graphing using derivatives
        • ▸Special points
          • ▸Finding critical values from graphs
            • •Lesson
            • •Practice
          • •Finding critical values using derivatives
          • ▸Inflection points
            • •Graphically finding points of inflection
            • •Using the second derivative to find points of inflection
          • •Classifying stationary points
          • •Sketching polynomials using stationary points
        • •Bisection method
        • •Newton's method
        • •False position method
        • •L'Hopital's rule
      • ▾Integrals
        • •Basic integration problems
        • ▸Integration by u-substitution
          • ▸Integration by u-substitution (less difficult)
            • •Lesson
            • •Practice
          • •Integration by u-substitution (more difficult)
          • •Integrating exponential functions by u-substitution
        • •Visually determining antiderivative
      • ▸Applications of integrals
        • •Proving the formula for the area of a circle
        • •Area between two curves
        • ▸Solids of revolution
          • •Disc and washer methods (circular cross sections)
          • •Volumes of solids with known cross sections
          • •Shell method
          • •Volume of a sphere (calculus \(2\))
          • •Volume of a cone (calculus 2)
     › Calculus 1 › Integrals

    Basic integration problems

    In this section, you'll do some basic integration problems. These problems will only involve elementary algebra and the application of basic rules of integration. These problems will not require u-substitution or integration by parts. Before attempting this section, you should be very good at performing elementary algebra, you should know your trig identities, and you should know basic integral identities, such as the reverse power rule.

    \(\displaystyle \int \left(1 + 2x - 4x^3\right)\,dx\)
    \(x + x^2 - x^4 + C\)
    70817❖ Basic Integration Problems
    patrickJMT
    \(\displaystyle \int \dfrac{3x - 2}{\sqrt{x}}\,dx\)
    \(2x^{3 / 2} - 4x^{1 / 2} + C\)
    70817❖ Basic Integration Problems
    patrickJMT
    \(\displaystyle \int \dfrac{\sin \theta + \sin \theta\tan^2 \theta}{\sec^2 \theta}\,d\theta\)
    \(-\cos \theta + C\)
    70817❖ Basic Integration Problems
    patrickJMT
    \(\displaystyle \int (x + 1)\left(x^2 + 3\right)\,dx\)
    \(\dfrac{x^4}{4} + \dfrac{3x^2}{2} + \dfrac{x^3}{3} + 3x + C\)
    70817❖ Basic Integration Problems
    patrickJMT
    \(\displaystyle\int x^2(3x - 1)\ dx\)
    \(\dfrac{3x^4}{4} - \dfrac{x^3}{3} + C\)
    172Rewriting before integrating
    Khan Academy ~ YouTube
    \(\displaystyle\int \dfrac{x^3 + 3x^2 - 5}{x^2}\ dx\)
    \(\dfrac{x^2}{2} + 3x + 5x^{-1} + C\)
    172Rewriting before integrating
    Khan Academy ~ YouTube
    \(\displaystyle\int \sqrt[3]{x^5}\ dx\)
    \(\dfrac{3}{8}x^{8 / 3} + C\)
    172Rewriting before integrating
    Khan Academy ~ YouTube

    Conclude by giving your students these challenges:

    • 2017 Math Kangaroo Levels 3-4 Problem #19 by STEM4all
    • More Children and Plants by NRICH
    • Three Four Five by NRICH
    • Going to the Cinema by NRICH
    • A Curious Surface Area

    YouTube videos

    • 400Basic Integration... How? (NancyPi)
      NancyPi
    • 4717Integral Constant Multiple Rule
      Firefly Lectures
    • 1396Review of Basic Integration Rules Calculus 1 AB - 6 Examples
      ProfRobBob