In this video, Krista King from integralCALC Academy shows how to find f(x), the original function, given f''(x), f double prime of x, or the second derivative of f, and initial conditions. Take the integral of f''(x) to get f'(x), then take the integral of f'(x) to get f(x). Make sure to add the constant of integration after each integration, then plug in both initial conditions to f(x). This will give you a system of equations that you can solve for both constants of integration.

1. Area Under the Curve (Example 1)
2. Area Under the Graph vs. Area Enclosed by the Graph
3. Summation Notation: Finding the Sum
4. Summation Notation: Expanding
5. Summation Notation: Collapsing
6. Riemann Sums Right Endpoints
7. Riemann Sums Midpoints
8. Trapezoidal Rule
9. Simpson's Approximation
10. Definite Integral
11. Definite Integral of an Even Function
12. Definite Integral of an Odd Function
13. Fundamental Theorem of Calculus (Part I)
14. Fundamental Theorem of Calculus (Part II)
15. Indefinite Integrals
16. Properties of integrals
17. Find f(x) Given f''(x), its Second Derivative
18. Find f Given f'' and Initial Conditions
19. Find f(x) Given f'''(x), its Third Derivative
20. Integral of a Quadratic Function
21. Initial Value Problem
22. U-Substitution
23. U-Substitution in Definite Integrals
24. U-Substitution with Integration by Parts
25. Integration by Parts
26. Integration by Parts Two Times
27. Integration by Parts Three Times
28. Integration by Parts to Prove the Reduction Formula
29. Tabular Integration
30. Partial Fractions, Distinct Linear Factors (Example 3)
31. Partial Fractions, Repeated Linear Factors
32. Partial Fractions, Distinct Quadratic Factors (Example 3)
33. Partial Fractions, Repeated Quadratic Factors
34. Partial Fractions, Rationalizing Substitution
35. Trigonometric Integrals
36. Trigonometric Integrals: sin^mcos^n and odd m
37. Trigonometric Integrals: sin^mcos^n and odd n
38. Trigonometric Integrals: sin^mcos^n, m and n even
39. Integrals of Trigonometric Functions: tan^msec^n and odd m
40. Integrals of Trigonometric Functions: tan^msec^n and even n
41. Integrals of Trigonometric Functions: sin(mx)cos(nx)
42. Integrals of Trigonometric Functions: sin(mx)sin(nx)
43. Integrals of Trigonometric Functions: cos(mx)cos(nx)
44. Integrals of Hyperbolic Functions
45. Integrals of Inverse Hyperbolic Functions
46. Setting Up Trigonometric Substitution
47. Trigonometric Substitution with Secant
48. Trigonometric Substitution with Tangent
49. Trigonometric Substitution with Sine
50. Improper Integral (Part I)
51. Improper Integral (Part II)
52. Integrals Using Reduction Formulas
53. Average Value of the Function
54. Area Between Curves - dx
55. Area Between Curves - dy
56. Area Between Curves: Sketching
57. Arc Length x=g(y)
58. Surface Area of Revolution
59. Surface of Revolution
60. Volume of Rotation: Disk Method about the y-xis
61. Volume of Rotation: Disk Method about the x-axis
62. Volume of Rotation: Washer Method about y-axis
63. Volume of Rotation: Washer Method about x-axis
64. Volume of Rotation: Cylindrical Shells about the y-axis
65. Volume of Rotation: Cylindrical Shells about the x-axis
66. Mean Value Theorem for Integrals
67. Work
68. Work Done on Elastic Springs
69. Work Done by a Variable Force
70. Center of Mass of the System
71. Moments of the System
72. Hydrostatic Force
73. Hydrostatic Pressure
74. Vertical Motion (Integration)
75. Rectilinear Motion
76. Centroids of Plane Regions
77. Area of the Triangle with the Given Vertices
78. Present and Future Value
79. Consumer and Producer Surplus
80. Probability Density Functions
81. Cardiac Output
82. Poiseuille's Law
83. Theorem of Pappus
84. Analytic Geometry: Graph of a Single Point or of No Points
85. Analytic Geometry: Set of Points Equally Distant from Two Points
86. Analytic Geometry: Set of Points Unequally Distant from Two Points
87. Eccentricity and Directrix of the Conic Section
88. Parabolas: Vertex, Axis, Focus, and Directrix
89. Equation of a Parabola (Conic Section)
90. Polar Equation of the Parabola (Conic Section)
91. Vertex Axis Focus Directrix of an Ellipse
92. Equation of an Ellipse (Conic Section)
93. Polar Equation of the Ellipse (Conic Section)
94. Vertex Axis Focus Directrix Asymptotes of a Hyperbola
95. Equation of a Hyperbola (Conic Section)
96. Polar Equation of the Hyperbola (Conic Section)
97. Eliminating the Parameter
98. Derivative of a Parametric Curve
99. Second Derivative of a Parametric Curve
100. Tangent Line to the Parametric Curve
101. Sketch the Parametric Curve by Plotting Points
102. Area Under the Parametric Curve
103. Parametric Area Under One Arc or Loop
104. Parametric Curve: Surface Area of Revolution
105. Surface Area of Revolution of a Parametric Curve Rotated About the y-axis
106. Parametric Arc Length
107. Parametric Arc Length and the distance Traveled by the Particle
108. Volume of Revolution of a Parametric Curve
109. Converting Polar Coordinates
110. Converting Rectangular Equations to Polar Equations
111. Converting Polar Equations to Rectangular Equations
112. Distance Between Two Polar Points
113. Sketching Polar Curves from Cartesian Curves
114. Sketching Polar Curves: 2 Examples
115. Tangent Line to the Polar Curve
116. Vertical and Horizontal Tangent Lines to the Polar Curve
117. Polar Area
118. Polar Area Bounded by One Loop
119. Points of Intersection of Two Polar Curves
120. Area Between Polar Curves
121. Polar Area Inside Both Curves
122. Arc Length of a Polar Curve
123. Polar Parametric Curve: Surface Area of Revolution
124. Polar Parametric Curve: Arc Length
125. Listing the First Terms of the Sequence
126. Calculating the First Terms of the Sequence
127. Finding a Formula for the General Term of the Sequence (a_n)
128. Does the Sequence Converge or Diverge?
129. Finding the Limit of a Convergent Sequence
130. Increasing, Decreasing and not Monotonic Sequences
131. Bounded Sequences
132. Calculating the First Terms in a Series of Partial Sums
133. Sum of the Series of Partial Sums
134. Repeating Decimal Expressed as a Ratio of Integers
135. nth Term Test, Divergence Test, and the Zero Test
136. Convergence of a Geometric Series
137. Convergence and Sum of a Geometric Series (Example 1)
138. Values for Which the Geometric Series Converges
139. Convergence of a Telescoping Series
140. Sum of Telescoping Series
141. p-Series Test for Convergence
142. Integral Test for Convergence
143. Comparison Test
144. Limit Comparison Test
145. Estimating Error/Remainder of a Series
146. Alternating Series Test
147. Alternating Series Estimation Theorem
148. Ratio Test
149. Ratio Test with Factorials
150. Root Test
151. Absolute and Conditional Convergence
152. Difference Between Limit and Sum of the Series
153. Radius of Convergence
154. Interval of Convergence
155. Power Series Representation, Radius and Interval of Convergence
156. Power Series Differentiation
157. Expressing the Integral as a Power Series
158. Using Power Series to Estimate a Definite Integral
159. Taylor Polynomial (Part I)
160. Taylor Polynomial (Part II)
161. Finding Radius of Convergence of a Taylor Series
162. Taylor's Inequality
163. Maclaurin Series
164. Sum of the Maclaurin Series
165. Maclaurin Series Radius of Convergence
166. Power Series Division
167. Power Series Multiplication
168. Binomial Series
169. Expressing an Indefinite Integral as an Infinite Series
170. Using Maclaurin Series to Estimate an Indefinite Integral
171. Maclaurin Series to Estimate a Definite Integral
172. Maclaurin Series to Evaluate a Limit
173. Improper Integrals (Case 2)
174. Partial Fractions: Two Ways to Find the Constants
175. Improper Integrals (Case 3)
176. Integrating with Partial Fractions: How to Factor Difficult Denominators

In this course, Krista King from the integralCALC Academy covers a range of topics in Calculus II, including Integrals, Applications of Integrals, Polar & Parametric of Sequences & Series.