Calculus 2 | mathXplain
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8 topics, 88 short and super clear episodes

This Calculus 2 course includes 88 short and super clear episodes that take you through 8 topics and help you navigate the bumpy roads of Calculus 2. The casual style makes you feel like you are discussing some simple issue, such as cooking scrambled eggs.

The course consists of 8 sections: Indefinite integral, Definite integral, Sequences, Infinite series, Power series, Fourier series, Differentiation of two variable function, Differential equations

## INTEGRATION

• Definite and indefinite integrals - Definite integration deals with calculating the area under the curve of a function. The indefinite integration works a totally different way. The indefinite integral of f(x) is a function that is called the primitive function.
• Primitive function - The symbol for the primitive function is F(x), and its main property is that if we differentiate it, we get f(x).
• Fundamental theorem of calculus - The theorem that describes this relationship is one of the most important theorems in the entire history of mathematics. It was developed at the end of the 1600s by two men at the same time: an English physicist named Newton and a German philosopher named Leibniz.
• Basic integration formulas - We start looking for primitive functions by remembering the derivatives of a few important functions.
• Inteon rulegratis - We will have to know which rule to use, and it is not recommended to decide by tossing a coin.
• Integration by parts - The purpose of partial integration is to turn a complicated integral into a simpler one.
• Integration of composite functions - This theorem is really about integrating composite functions. But unfortunately composite functions are a bit problematic, as computing their integral is usually a quite hopeless undertaking. Let's see.
• Integration by substitution - The main idea of integration by substitution is that we replace an expression by u, hoping that this way we may be able to solve the problem.
• Partial fractions - Integrating rational functions is an extremely entertaining activity. We will start by developing our ability to integrate so called partial fractions.
• Integrating rational functions - We can easily integrate any rational function. All we have to do is decompose it into partial fractions, and then integrate the partial fractions.
• Polynomial division - This polynomial division is just like long division we learned in elementary school, but that much funnier.
• Integration of trigonometric functions - Integration of trigonometric terms is not easy. We will check out a few simpler tricks and the most important methods.
• Tangent half-angle substitution - One of the strangest cases of integration by substitution is when u= tan(x/2). We use this if sinx and cosx is included in first-degree form in the fraction.
• Definite integral - Definite integration deals with calculating the area under the curve of a function.
• Improper integrals - These integrals that stretch into infinity are called improper integrals.

## SEQUENCES

• Index of sequences - The index of the sequence tells us which member of the sequence we are looking at.
• Limits of sequences - One of the most important properties of a sequence is what happens to it when we look at its members that are further and further away.
• Common sequences - Exponential, polynomial and radical sequences.
• Limits and operations - Well, the point of these tables is to help us navigate among the various types of limits.
• Find the limits - Divide both the numerator and the denominator by the strongest member of the denominator.
• Convergent sequences - A sequence is called convergent if there is a real number that is the limit of the sequence.
• Divergent sequences - A sequence is divergent if it tends to infinity, but it is also divergent if it doesn’t tend to anywhere at all.
• Oscillating sequences - Sequences that tend to nowhere are always oscillating sequences.
• Squeeze Theorem - We need a sequence that is less than the original sequence, and another one that is greater than the original sequence.
• Estimation - When finding the lower estimate, we omit everything but the strongest term, and when finding the upper estimate, we replace all terms with the strongest term.

## INFINITE SERIES

• Series - A sum that we get by adding up an infinite sequence of real numbers is called an infinite series.
• Geometric series - Let's see how we compute the sum of geometric series.
• Sequence of partial sums - The sequence of partial sums tells us where the flea is at the moment.
• Convergent series - An infinite series is convergent if the associated sequence of partial sums (Sn) is convergent.
• Convergence tests - nth-term test for divergence, Root test, Ratio test, Direct Comparison Test. Let’s see how we can determine whether a series is convergent or divergent.
• Leibniz-series - Confused weakening fleas, on the other hand, are guaranteed to converge. This was discovered by Leibniz.
• Direct comparison test - this test to distinguish it from similar related tests, provides a way of deducing the convergence or divergence of an infinite series. In both cases, the test works by comparing the given series or integral to one whose convergence properties are known..
• Telescoping series - It is time to compute the sum of another special series, the so called telescoping series.

## DIFFERENTIAL EQUATIONS

• Differential equations - Differential equations are equations where the unknowns are functions. The equations contain various derivatives and exponentials of these functions.
• Order of equations - It indicates the highest derivative of the function in the equation.
• Linearity - If the equation includes the unknown function and its derivatives only in first-degree form, then the equation is linear.
• Ordinary differential equation - If the differential equation contains a function of one variable, then it is called an ordinary differential equation (ODE);
• Separable differential equations - Move all y terms to the dy side, and move all x terms to the dx side, and integrate both sides to get the solution.
• General solution - We get the general solution if we give C a arbitrary value.
• Particular solution - We get the particular solution if we give C a fix value.
• Exact differential equations - The p(x,y)dx+q(x,y)dy=0 equation is exact, if there exists a function F(x,y) such that F’x=p and F’y=q, then, the solution for the equation is that function F(x,y)=C
• Integrating factor - If the equation is not exact, then we try to make it exact by using an integrating factor.
• First-order linear differential equations - In general, a first-order linear differential equation contains a y’ and it also contains a first-order y.
• First-order constatnt coeffitient linear differential equations - This type is a special case of the first-order linear differential equations.
• First order homogeneous equation - The homogeneous equation is y’+ay=0
• Homogeneous solution - The homogeneous solution is the solution of homogeneous equation.
• Particular solution - We can obtain the particular solution based on the function on the right side, using a very funny procedure called “Method of Undetermined Coefficients”, or “Trial Functions Method”.
• Trial Functions Method - We can obtain the particular solution based on the function on the right side, using a very funny procedure called “Method of Undetermined Coefficients”, or “Trial Functions Method”.
• Method of Undetermined Coeffitients - We can obtain the particular solution based on the function on the right side, using a very funny procedure called “Method of Undetermined Coefficients”, or “Trial Functions Method”.
• Resonance in first order cases - This occurs when there is eAx in the particular solution, and its exponent is exactly the same as the exponent of the homogeneous solution.
• Second order homogeneous equation - The homogeneous equation is ay"+by'+cy=0
• Homogeneous solution - The homogeneous solution is the solution of homogeneous equation.
• Characteristic equation - In order to solve the homogeneous equation, we have to solve this quadratic equation.
• Resonance in second order cases - One term of the homogeneous solution is equal to a term of the particular solution. That means that sadly, there is resonance. When the characteristic equation has only one real solution, there could be dual resonance.