Better Explanation

Exam #3 Formula Sheet

Indefinite Integration

• Variable of Integration: denoted typically by $x$.
• Constant of Integration: represented by $C$.
• Antidifferentiation Formula: For a function $y = f(x)$, the indefinite integral is expressed as:
  • $y = ext{indefinite integral of } f(x) ext{dx} = F(x) + C$
  • Where $F(x)$ is an antiderivative of $f(x)$.

Steps for Finding Indefinite Integral:

1. Rewrite Functions: If necessary, transform square roots and powers; for instance:

   • $ext{Square roots: } ext{e.g., } ext{ } ext{ } ext{sq. root} = x^{1/2}$
   • $ext{From: } x^{-n}$ where $n = 1, 2, …$

2. Apply Power Rule: For any power of $x$, integrate using:

   • $ext{If } f(x) = x^n, ext{ then } ext{integral: } F(x) = \frac{x^{n+1}}{n+1} + C$

3. Combine Results: Sum the results from each application of the power rule and add the constant $C$.

   • Example: If integrating $x^2$, we get:
   • $ext{Integrate: } ext{x}^2 = \frac{x^{2+1}}{2+1} + C = \frac{x^3}{3} + C$

4. Using Trigonometric Functions: Find the corresponding integration formula and ensure to place coefficients after any negative signs.

Finding Particular Solutions of Differential Equations

Steps for Solving:

1. Set the Antiderivative: Establish that $F(x) = F'(x)$ according to any initial conditions.

2. Apply Power & Constant Rules: Remember to include the constant of integration.

3. Substituting Known Values: Insert any known values of $x$ and $F(x)$ to find $C$.

4. Obtain Full Solution: Substitute the value of $C$ back into the equation to complete the solution.

Solving for Height Over Time (Differential Equations)

Steps to Follow:

1. Integrate Both Sides: Perform integration with respect to $t$:

   • $h(t) = ext{integral}(h) = ext{integral}\bigg(\frac{d(wh)}{dt}\bigg)dt$

2. Apply Power Rule: Add variables to all coefficients, don't forget to include $C$.

3. Determine Initial Height: Check what the height was at the initial time, marking down its value.

4. Substituting Initial Values: Insert the initial time into the equation $h(t)$ to resolve for $C$.

5. Full Particular Solution: Integrate the result to obtain the full solution with all constants.

Calculating Sums with Summation Concepts

Principles to Follow:

1. Breaking Down Each Term: Start from the initial term $i$, progressively integrate until completion.

2. Combine Results: Add all calculated answers together for the final summation.

3. Using Summation Properties:

   • Simplify Summation Expressions: Expand any necessary terms before substituting back into the equation.
   • When $k$ is constant: Use properties like:

   - ext{For $orall k$, } extstyle ext{ } ext{Sum } igg( extstyle ext{ } ext{ } ext{ } k imes ai igg) = k imes ext{Sum}(ai)

   • Apply Summation Formulas: For constant $C$ use:
   • $ext{Sum} \big( ext{C} \big) = Cn$
   • Ensure to separate terms using appropriate expressions for $n(n+1)/2$.

4. Calculate Final Sum: Evaluate based on part one, part two, and part three components.

Evaluating Definite Integral by Limit Definition

Steps to Evaluate:

1. Set Up Limit Definition:

   • $ext{Definite Integral: } ext{ } ext{Integral}(f(x)) ext{dx} = ext{lim} ext{ as } n o ext{infinity} ext{ sum of } f(c_i) imes ext{A}x$

2. Calculate $ ext{A}x$: Determine the width of your intervals, calculating necessary values accordingly.

3. Determine Value of Function: Substitute $c_i$ values back into original $f(x)$ to ascertain outputs as needed.

4. Riemann Sum: Find the limit concisely expressed as:

   • $ext{Sum} f(C<em>i) imes A</em>x$

5. Calculate Limit: Execute the limit on the summation results finally adjusting to eliminate $n$ terms.

Fundamental Theorem of Calculus (FTC)

• Faster Evaluations: The Fundamental Theorem allows faster computation.
  • ext{For every } f(u) ext{, integral from } a ext{ to } b $ ext{ is } :
  • $ext {Evaluate } F(b) - F(a)$

Rate of Change Over Time

Evaluate Using the Definite Integral

1. Set Up the Definite Integral: Integrate over the interval determined by the initial time $a$ to final time $b$:

   • $ext{Evaluate: } ext{Integral}(R(t)) ext{dt where } t = a ext{ to } t = b$

2. Find Antiderivative: Calculate the antiderivative over the specified interval altering expressions accordingly.

3. Use FTC to Find Outputs: Determine final output via:

   • $F(b) - F(a)$

Integration Techniques with Change of Variables

Finding Indefinite Integrals:

1. Identify Substitution (u = g(x)): Choose $u$ such that it simplifies integration. Typically, the most complex part of the function, such as the highest exponent.

2. Deriving for $du$: Compute $du$ ensuring it matches any additional $x$ terms as required.

   • Importance of Balance: If the $du$ is a constant, divide by that coefficient.

3. Substituting u and du: Use the chosen $u$ back into the integral after established.

4. Apply Power Rule to u: Integrate $u$, however, do not include $C$ when involving u but consider limits again for symmetry.

5. Final Substitution: Substitute your original variable back in the computed expression.