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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 › Integration by u-substitution

    Integration by u-substitution (more difficult)

    In this section, you will practice integration by u-substitution on some slightly harder problems. These problems will require a careful choice of \(u\) in order to remove all occurences of \(x.\) Before attempting the problems in this section, you should know how to integrate using u-substition for problems where the choice of \(u\) is obvious.

    \(\displaystyle \int e^{x + e^x}\,dx\)
    \(\begin{align} & \int e^{x + e^x}\,dx \\[0.4em] =\ & \int e^x \cdot e^{e^x}\,dx \\[0.4em] =\ & \int e^u\,du \\[0.4em] =\ & e^u + C \\[0.4em] =\ & e^{e^x} + C \end{align}\) \(\begin{align} & u = e^x \\ & du = e^x\,dx \end{align}\)
    70834U-Substitution - More Complicated Examples
    patrickJMT
    \(\displaystyle \int \cot x[\ln(\sin x)]\,dx\)
    \(\begin{align} & \int \cot x[\ln(\sin x)]\,dx \\[0.4em] =\ & \int u\,du \\[0.4em] =\ & \dfrac{u^2}{2} + C \\[0.4em] =\ & \dfrac{[\ln(\sin x)]^2}{2} + C \end{align}\) \(\begin{align} & u = \ln(\sin x) \\[0.4em] & du = \dfrac{1}{\sin x}\cos x\,dx \\[0.4em] & du = \cot x\,dx \end{align}\)
    70834U-Substitution - More Complicated Examples
    patrickJMT
    \(\displaystyle \int \dfrac{x}{1 + x^4}\,dx\)
    \(\begin{align} & \int \dfrac{x}{1 + x^4}\,dx \\[0.75em] =\ & \int \dfrac{x}{1 + \left(x^2\right)^2}\,dx \\[0.75em] =\ & \int \dfrac{x}{1 + u^2} \cdot \dfrac{du}{2x} \\[0.75em] =\ & \int \dfrac{1}{1 + u^2} \cdot \dfrac{du}{2} \\[0.75em] =\ & \dfrac{1}{2} \int \dfrac{1}{1 + u^2}\,du \\[0.75em] =\ & \dfrac{1}{2}\tan^{-1} u + C \\[0.75em] =\ & \dfrac{1}{2}\tan^{-1}\left(x^2\right) + C \end{align}\) \(\begin{align} & u = x^2 \\ & du = 2x\,dx \end{align}\)
    51278understand u substitution, 3 slightly harder examples, (expo 2-in-1 marker review)
    blackpenredpen
    \(\displaystyle \int \tan x\ln(\cos x)\,dx\)
    \(\begin{align} & \int \tan x\ln(\cos x)\,dx \\[0.75em] =\ & \int (\tan x) u\dfrac{du}{-\tan x} \\[0.75em] =\ & -\int u\,du \\[0.75em] =\ & \dfrac{-1}{2}u^2 + C \\[0.75em] =\ & \dfrac{-1}{2}[\ln(\cos x)]^2 + C \end{align}\) \(\begin{align} & u = \ln(\cos x) \\[0.4em] & du = \dfrac{1}{\cos x} \cdot (-\sin x)\,dx \\[0.4em] & du = -\tan x\,dx \\[0.4em] & \dfrac{du}{-\tan x} = dx \end{align}\)
    51278understand u substitution, 3 slightly harder examples, (expo 2-in-1 marker review)
    blackpenredpen
    \(\displaystyle \int \dfrac{1}{1 + \sqrt{x}}\,dx\)
    \(\begin{align} & \int \dfrac{1}{1 + \sqrt{x}}\,dx \\[0.75em] =\ & \int \dfrac{1}{u}2(u - 1)\,du \\[0.75em] =\ & 2\int \dfrac{u - 1}{u}\,du \\[0.75em] =\ & 2\int \dfrac{u}{u} - \dfrac{1}{u}\,du \\[0.75em] =\ & 2\int 1 - \dfrac{1}{u}\,du \\[0.75em] =\ & 2(u - \ln \lvert u \rvert + C) \\[0.75em] =\ & 2u - 2\ln \lvert u \rvert + C \\[0.75em] =\ & 2(1 + \sqrt{x}) - 2\ln \lvert 1 + \sqrt{x} \rvert + C \\[0.75em] =\ & 2\sqrt{x} - 2\ln(1 + \sqrt{x}) + C \end{align}\) \(\begin{align} u = 1 + \sqrt{x} \\[0.4em] u - 1 = \sqrt{x} \\[0.4em] du = \dfrac{1}{2\sqrt{x}}\,dx \\[0.4em] 2(u - 1)\,du = dx \end{align}\)
    51278understand u substitution, 3 slightly harder examples, (expo 2-in-1 marker review)
    blackpenredpen
    \(\displaystyle \int \dfrac{e^x + 1}{e^x}\,dx\)
    \(x - e^{-x} + C\)
    23556Integrating Exponential Functions - Examples 1 and 2
    patrickJMT
    \(\displaystyle \int 3^{\sin \theta}\cos \theta\,d\theta\)
    \(\dfrac{3^{\sin \theta}}{\ln 3} + C\)
    23556Integrating Exponential Functions - Examples 1 and 2
    patrickJMT

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