4.2 Integration

• What is the reverse process of differentiation called?

  Integration
• The power rule for integration states that \int x^{n} dx = \frac{x^{n + 1}}{n + 1} + C</latex>, where $n \neq - 1$ and $C$ is the constant of integration
• The constant of integration, $C$, is necessary in indefinite integrals.
• What is the result of integrating $x^{3}$ with respect to $x$?


  $\frac{x^{4}}{4} +$$C$
• The constant rule for integration states that $\int k dx =$$kx +$$C$, where $k$ is a constant
• What is the result of integrating $5$ with respect to $x$?


  $5x +$$C$
• The constant multiple rule states that $\int k f(x) dx =$$k \int f(x) dx$, where $k$ is a constant.
• What is the result of integrating $2x$ with respect to $x$?


  $x^{2} +$$C$
• The sum/difference rule for integration states that $\int [f(x) \pm g(x)] dx =$$\int f(x) dx \pm \int g(x) dx$, which allows you to split the integral into separate terms
• What is the result of integrating $x^{2} +$$3x$ with respect to $x$?


  $\frac{x^{3}}{3} +$$\frac{3x^{2}}{2} +$$C$
• A definite integral represents the area under a curve between two limits.
• The fundamental theorem of calculus states that \int_{a}^{b} f(x) dx = F(b) - F(a)</latex>, where $F(x)$ is the antiderivative of $f(x)$.
• What is the value of $\int_{1}^{3} (2x + 1) dx$?


  $10$
• Steps to evaluate a definite integral using the fundamental theorem of calculus

  1️⃣ Find the antiderivative of the function

  2️⃣ Evaluate the antiderivative at the upper and lower limits

  3️⃣ Subtract the lower limit value from the upper limit value
• What should you always include when finding an indefinite integral?

  $+$$C$
• A definite integral has numerical limits, while an indefinite integral does not.
• Integration by substitution is a technique used to simplify integrals by changing the variable of integration
• In integration by substitution, what do you typically substitute $u$ for?


  $g(x)$
• Steps to perform integration by substitution

  1️⃣ Identify a suitable substitution $u =$$g(x)$

  2️⃣ Compute $du =$$g'(x) dx$

  3️⃣ Rewrite the integral in terms of $u$

  4️⃣ Evaluate the new integral

  5️⃣ Substitute back to express the result in terms of $x$
• Integration by substitution simplifies integrals by changing the variable of integration
• Integration by substitution is useful when the integrand contains a composite function.
• Steps to integrate by substitution

  1️⃣ Identify a suitable substitution $u =$$g(x)$

  2️⃣ Compute the derivative $du =$$g'(x) dx$

  3️⃣ Express the integral in terms of $u$

  4️⃣ Evaluate the new integral

  5️⃣ Substitute back to express the result in terms of $x$
• The derivative $g'(x)$ must be present in the integrand for integration by substitution to work.
• Match each step of integration by substitution with its description:

  Choose a substitution $u =$$g(x)$ ↔️ Identify a composite function

  Calculate $du =$$g'(x) dx$ ↔️ Compute the derivative

  Rewrite integral in terms of $u$ ↔️ Change the variable

  Integrate with respect to $u$ ↔️ Evaluate the new integral
• Evaluate $\int 2x(x^{2} +$$3)^{4} dx$ using integration by substitution
• Steps to evaluate \int 2x(x^{2} + 3)^{4} dx</latex> using integration by substitution

  1️⃣ Let $u =$$x^{2} +$$3$

  2️⃣ $du =$$2x dx$

  3️⃣ Rewrite the integral: $\int u^{4} du$

  4️⃣ Integrate: $\frac{u^{5}}{5} +$$C$

  5️⃣ Substitute back: \frac{(x^{2} + 3)^{5}}{5} +$C$
• Integration by substitution simplifies complex integrals
• The formula for integration by parts is $\int u dv =$$uv - \int v du$, where $u$ and $dv$ are parts of the original integral
• Steps to apply integration by parts

  1️⃣ Identify $u$ and $dv$ using LIATE

  2️⃣ Calculate $du$ and $v$

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

  4️⃣ Evaluate the new integral

  5️⃣ Simplify
• Match each step of integration by parts with its description:

  Identify $u$ and $dv$ ↔️ Use LIATE to choose

  Calculate $du$ and $v$ ↔️ Compute the derivatives and integrals

  Apply the formula ↔️ Rewrite the integral

  Evaluate the new integral ↔️ Solve the remaining integral
• Integration by parts is used for products of functions
• When using integration by parts, chooseu</latex> to simplify when differentiated.
• Match each trigonometric function with its integral:

  $\sin x$ ↔️ $- \cos x +$$C$

  $\cos x$ ↔️ $\sin x +$$C$

  $\sec^{2} x$ ↔️ $\tan x +$$C$

  $\tan x$ ↔️ $- \ln |\cos x| +$$C$
• The power rule for integration states that $\int x^{n} dx =$$\frac{x^{n + 1}}{n + 1} +$$C$, where $n \neq - 1$ and $C$ is the constant of integration
• The integral of a constant $k$ is $kx +$$C$.
• The constant multiple rule allows you to move a constant outside the integral.
• What is integration the reverse process of?

  Differentiation
• $\int k dx =$$kx +$$C$, where $k$ is a constant.
• $\int k f(x) dx =$$k \int f(x) dx$, where $k$ is a constant</latex>
• $\int [f(x) \pm g(x)] dx =$$\int f(x) dx \pm \int g(x) dx$