6.14 Selecting Techniques for Antidifferentiation

• What is another name for antidifferentiation?

  Integration
• The formula for antidifferentiation is \int f'(x) \, dx = f(x) + C</latex>
• Antidifferentiation is vital in calculus for solving differential equations
• Integration by u-substitution transforms complex functions into simpler, more manageable forms
• U-substitution is used when the integrand contains a function and its derivative.
• Steps to perform u-substitution:

  1️⃣ Identify a function and its derivative

  2️⃣ Express the integrand in terms of u

  3️⃣ Evaluate the integral

  4️⃣ Back-substitute to express the result in terms of x
• What is the result of $\int 2x \sqrt{1 + x^{2}} \, dx$?


  $\frac{2}{3} (1 + x^{2})^{3 / 2} +$$C$
• Match the basic integration techniques with their formulas:

  Power Rule ↔️ $\int x^{n} \, dx =$$\frac{x^{n + 1}}{n + 1} +$$C$

  Constant Multiple Rule ↔️ $\int kf(x) \, dx =$$k \int f(x) \, dx$

  Sum and Difference Rules ↔️ $\int [f(x) \pm g(x)] \, dx =$$\int f(x) \, dx \pm \int g(x) \, dx$

  Simple Trigonometric Integrals ↔️ $\int \sin x \, dx =$$- \cos x +$$C$, $\int \cos x \, dx =$$\sin x +$$C$

  Exponential Integrals ↔️ $\int e^{x} \, dx =$$e^{x} +$$C$
• U-substitution is used when the integrand involves composition of functions.
• Steps to perform u-substitution:

  1️⃣ Choose u

  2️⃣ Calculate du

  3️⃣ Substitute u and du into the integral

  4️⃣ Integrate with respect to u

  5️⃣ Back-substitute to express the result in terms of x
• What is the formula for integration by parts?

  $\int u \, dv =$$uv - \int v \, du$
• Evaluate $\int x \sin x \, dx$ using integration by parts
• In integration by parts, if $u =$$x$, then $du =$$dx$.
• Match the expression under the square root with the correct trigonometric substitution:

  $a^{2} - x^{2}$ ↔️ $x =$$a \sin \theta$

  $a^{2} +$$x^{2}$ ↔️ $x =$$a \tan \theta$

  $x^{2} - a^{2}$ ↔️ $x =$$a \sec \theta$
• Evaluate $\int \frac{dx}{\sqrt{16 - x^{2}}}$ using trigonometric substitution
• What is the result of $\int \frac{dx}{\sqrt{16 - x^{2}}}$?


  \arcsin \frac{x}{4} + C</latex>
• U-substitution and trigonometric substitution are both used to simplify complex integrals.
• What type of substitution simplifies integrals with square root expressions using trigonometric identities?

  Trigonometric substitution
• Antidifferentiation is also known as integration
• The constant of integration in antidifferentiation is denoted as $C$.
• What is one use of antidifferentiation in calculus?

  Calculate areas under curves
• The mathematical representation of antidifferentiation includes the constant of integration
• What are the basic integration techniques used in calculus?

  Power rule, constant multiple rule, sum and difference rules, trigonometric integrals, exponential integrals
• Match the integration technique with its description:

  Power Rule ↔️ Integrates $x^{n}$

  Constant Multiple Rule ↔️ Factors out constants

  Sum and Difference Rules ↔️ Integrates sums/differences

  Simple Trigonometric Integrals ↔️ Integrates sine and cosine
• The power rule states that $\int x^{n} \, dx =$$\frac{x^{n + 1}}{n + 1} +$$C$.
• What does the constant multiple rule allow you to do when integrating a function?

  Factor out constants
• The integral of $\cos x$ is \sin x
• What is the integral of $e^{x}$?


  $e^{x} +$$C$
• Basic integration techniques are crucial for calculating areas under curves.
• What is the purpose of integration by u-substitution?

  Simplify integrals
• U-substitution is used by changing the variable of integration
• Steps to perform integration by u-substitution:

  1️⃣ Choose u such that $g'(x)$ is in the integrand

  2️⃣ Calculate $du =$$g'(x) dx$

  3️⃣ Substitute $u$ and $du$ into the integral

  4️⃣ Evaluate $\int f(u) \, du$

  5️⃣ Back-substitute $u =$$g(x)$
• The integral of $2x \sqrt{1 + x^{2}} \, dx$ is $\frac{2}{3} (1 + x^{2})^{3 / 2} +$$C$.
• When should u-substitution be used instead of basic integration techniques?

  With composite functions
• Integration by parts is used to simplify integrals involving products of functions.
• Find $\frac{du}{dx} =$$g'(x)$ and $du =$$g'(x) \, dx$ to use in u-substitution
• Steps of u-substitution in integration

  1️⃣ Choose $u =$$g(x)$

  2️⃣ Calculate $du =$$g'(x) \, dx$

  3️⃣ Substitute $u$ and $du$ into the integral

  4️⃣ Evaluate $\int f(u) \, du$

  5️⃣ Back-substitute $u$ with $g(x)$
• U-substitution is used when basic integration techniques fail, particularly with composite functions and integrands involving a function and its derivative.
• The formula for integration by parts is \int u \, dv = uv - \int v \, du</latex>, where `u` and `dv` are chosen from the integrand
• Steps to apply integration by parts

  1️⃣ Choose $u$ and $dv$

  2️⃣ Find $du$ and $v$

  3️⃣ Apply the formula $\int u \, dv =$$uv - \int v \, du$