# Calculus with analytic geometry II

## Contents

1 Integration by parts

2 Trigonometric integrals

3 Trigonometric substitution

4 Partial fractions

5 Summary of integration techniques

6 Improper integrals

7 Separable differential equations

# Topics Links to video
0.0 Welcome video

## Integration by parts

# Topics Links to video
1 Notes on integration by parts
1.1 Integration by parts: motivation and formula
1.2 Integrating $$x \sin x$$ and the choice of $$u$$
1.3 How to not integrate $$x \sin x$$

Do not repeat at home
1.4 Getting rid of the second power of $$t$$ by integrating by parts

Do repeat at home (twice)
1.5 Integrating inverse functions and definite integrals by parts
1.6 Integration by parts with moving terms to the other side

In this example, we repeatedly integrate by parts and move some of the terms around to arrive at the answer
1.7 We obtain the reduction formula for $$\sin x$$

This example shows how trig identities will be useful for integration, and prepares us for the next section

## Trigonometric integrals

# Topics Links to video
2 Notes on trigonometric integrals
2.1 What is a trigonometric integral
2.2 We integrate $$\sin^5 x \, \cos^2 x$$

What is common to these products so far?
2.3 Applying half-angle formulas

After realizing that the previous examples contain odd degrees of $$\sin, \cos$$, we discuss dealing with even powers
2.4 In order to integrate $$\sin^4 x$$, half-angle formulas have to be applied repeatedly
2.5 The above examples are summarized into a general strategy for integrating products of $$\sin x$$ and $$\cos x$$
2.6 What is essential about our integration strategy? Why does it work?
2.7 Examples of integrating products of $$\tan x$$ and $$\sec x$$
2.8 A general strategy for products of $$\tan x$$ and $$\sec x$$ is obtained in this video
2.9 Some special cases not covered by the strategy for $$\tan x$$ and $$\sec x$$

Some products of $$\tan x$$ and $$\sec x$$ are still not covered by our general strategy. This video discusses how to use the Pythagorean identity and integration by parts to obtain reduction formulas for such cases, not covered previously.

## Trigonometric substitution

# Topics Links to video
3 Notes on trigonometric substitution
3.1 Motivation for trig sub

Evidence that trig sub shows up in some natural integrals.
3.2 Main table of trigonometric substitutions

We summarize the motivational examples into a table of different trig subs and identities that explain them.
3.3 Substituting $$x = a \sin\theta$$
3.4 Substituting $$x = a \sec\theta$$
3.5 Definite integrals with trig sub and handling coefficients on $$x^2$$

Instead of $$\pm x^2 \pm a^2$$ under the radical, in this example we have $$\pm c^2 x^2 \pm a^2$$. We show how to factor out the coefficient at $$x^2$$ from the integral.
3.6 Completing the square to apply a trig sub

In this example, expression under the square root is not of the form $$\pm x^2 \pm a^2$$, but instead a complete quadratic polynomial. We show how to reduce it to the standard form.

## Partial fractions

# Topics Links to video
4 Notes on partial fraction decomposition
4.1 Motivation for partial fraction decomposition

Definition of a rational function and the general idea of partial fraction decomposition
4.2 Main table of partial fraction decomposition

An overview of the algorithm for partial fraction decomposition and a table for the partial fractions corresponding to different irreducible factors in denominator.
4.3 Long division and unique linear factors

We use long division to split an improper rational function into a sum of a polynomial with a proper part. In the second example, we integrate a rational function with unique (nonrepeated) factors in the denominator.
4.4 Repeated linear factors

An example of a rational function with repeated linear factors in the denominator. We determine the coefficient by equating the coefficients for the powers of $$x$$.
4.5 Unique quadratic factors

Integration of a rational function with unique (nonrepeated) irreducible quadratic terms.
4.6 Repeated quadratic factors

Integration of functions with repeated irreducible quadratic terms. In particular, we see how to apply trig sub to negative powers of $$1+x^2$$.
4.7 Completing a square in a partial fraction

We demonstrate how to integrate a partial fraction with an irreducible quadratic factor by completing a square, and applying a u-sub.
4.8 Rationalizing substitution

When integrating an expression with an n-th order root, we introduce a u-sub equal to this root, and reduce the problem to integration of a rational function.

## Summary of integration techniques

# Topics Links to video
5 Notes on the integration strategy
5.1 An overview of integration heuristics
5.2 Examples 1-2

These examples illustrate simplification of the integrand before integration and combining two different techniques: u-sub and integration by parts.
5.3 Examples 3-5

A discussion of the PFD and the usefulness of algebraic transformations, such as multiplying by the conjugate expression.
5.4 Conclusion: not all expressions have antiderivatives in elementary functions

Our course just scratches the surface of the vast functional universe. The antiderivatives of some of its inhabitants can only be expressed through special functions (error function, Bessel functions, etc), others don’t have a nice representation at all.

## Improper integrals

# Topics Links to video
6 Notes on improper integrals
6.1 Definition of an improper integral
6.2 Convergence of the integral of $$1/x^p$$ for different values of $$p$$
6.3 Using integration by parts in improper integrals
6.4 Improper integrals of type II and the comparison theorem

Improper integrals of the second type contain a function that becomes infinite at some point of the integration interval. We also discuss the comparison theorem and see its application to integral of $$e^{-x^2}$$. Specifically, even though this function does not have an antiderivative in elementary functions, we are able to determine that its improper integral converges.

## Separable differential equations

# Topics Links to video
7 Notes on differential equations
7.1 Introduction: why we care about integration techniques and an outline of the solution process
7.2 Applying the initial condition to select a specific solution
7.3 When solving for $$y$$ can be tricky
7.4 Absorb $$\pm$$ into the positive constant, by extending the range of admissible constants

This is useful when solutions have the form $$|y| = K e^x, \ K >0$$.
7.5 Modeling a simple electric circuit
7.6 Mixing problems

A common class of problems which involve uniform mixing of a solution.
7.7 Orthogonal trajectories

These generalize both Cartesian and polar coordinates. Imagine curved coordinate lines!