周次: 5 日期: March 25, 2022 节次: 2

In this course, we will discuss the definition of hyperbolic functions. And we shift our attention to several integration techniques. These techniques can help us integrate more functions. A typical technique called “Integration by Parts” will be discussed as well.

§ 7.8 A Brief Introduction on Hyperbolic Functions

Let’s recall the odd and even functions. We have stated that for every function $f$ defined on an interval centered at the origin can be written in a unique way as the sum of one even function and one odd function. The decomposition is

$$ f(x) = \underbrace{\frac{f(x) + f(-x)}{2}}{\text{even part}} + \underbrace{\frac{f(x)-f(-x)}{2}}{\text{odd part}} $$

No exception for exponential function $e^x$ , it can be decomposed. And we assigned two new functions to the decomposition.

$$ \cosh(x) := \text{even part of }e^x = \frac{e^x + e^{-x}}{2} $$

$$ \sinh(x) := \text{odd part of }e^x = \frac{e^x - e^{-x}}{2} $$

$\cosh(x)$ is pronounced as “kosh $x$”, rhyming with “gosh $x$”, and $\sinh(x)$ is pronounced as “cinch $x$”, rhyming with “pinch $x$”.

Graphs of Hyperbolic Functions

Identities for Hyperbolic Functions

$$ \cosh^2(x) - \sinh^2(x) = 1 $$

$$ \sinh(2x) = 2\sinh(x)\cosh(x) $$

$$ \cosh(2x) = \cosh^2(x) + \sinh^2(x) $$

Other Hyperbolic Functions

Analogous to trigonometric functions, the $\tanh(x)$, $\coth(x)$, $\mathrm{sech}(x)$ and $\mathrm{csch}(x)$ can all be defined, following the same pattern.

$$ \tanh(x) :=\dfrac{\sinh(x)}{\cosh(x)},\quad \coth(x):=\dfrac{\cosh(x)}{\sinh(x)}, \quad \text{etc.} $$

The Derivative and Anti-derivative of Hyperbolic Functions

$$ \frac{d}{dx}\sinh(x) = \cosh(x),\quad \frac{d}{dx}\cosh(x) = \sinh(x) $$

Therefore the anti-derivatives read

$$ \int\sinh(x) = \cosh(x) + C,\quad \int\cosh(x) = \sinh(x) + C. $$

Chapter 8. Techniques of Integration

In this chapter we study a number of important techniques for finding indefinite integrals of more complicated functions

§ 8.1 Basic Integration Formulas

Try to match integrals that confront us against one of the standard types. This usually involves a certain amount of algebraic manipulation as well as use of the substitution rules.

Recall that the substitution rule is

$$ \int f(g(x))g'(x)dx = \int f(u) du, $$

where $u=g(x)$ is a differentiable function whose range is an interval $I$ and $f$ is continuous on $I.$ Success in integration often hinges on the ability to spot what part of the integrand should be called $u$ in order that one will also have $du$, so that a known formula can be applied. This means that the first requirement for skill in integration is a thorough mastery of the formulas for differentiation.

**Note** Students should have a good mastery of formulas 12, 13, 18, 19. For 20 and 22, we provide an alternative (equivalent) anti-derivative

$$ \int\dfrac{du}{\sqrt{u^2\pm a^2}} = \ln |u+\sqrt{u^2\pm a^2}| + C $$

For 20, we have an alternative way to show the anti-derivative, by applying the integration table.

$$ \int\dfrac{du}{u\sqrt{u^2-a^2}} = \int \dfrac{du}{u^2\sqrt{1-\left(\frac{a}{u}\right)^2}}=-\frac1a\int \dfrac{d\left(\frac{a}u\right)}{\sqrt{1-\left(\frac{a}{u}\right)^2}}=-\frac1a \arcsin\left(\frac{a}{u}\right) + C $$

Example 1. Making a Simplifying Substitution

Evaluate

$$ \int\dfrac{2x-9}{\sqrt{x^2-9x+1}}dx $$

Example 2. Completing the Square

Evaluate

$$ \int\dfrac{dx}{\sqrt{8x-x^2}}dx $$

Example 3. Expanding a Power and Using a Trigonometric Identity

Evaluate

$$ \int (\sec x + \tan x)^2 dx. $$

Example 4. Eliminating a Square Root

Evaluate

$$ \int_0^{\pi/4} \sqrt{1+\cos 4x} ,dx. $$

Example 5. Reducing an Improper Fraction

Evaluate

$$ \int \dfrac{3x^2-7x}{3x+2},dx. $$

Example 6. Separating a Fraction

Evaluate

$$ \int \dfrac{3x+2}{\sqrt{1-x^2}},dx. $$

Example 7. Integral of $y=\sec(x)$ — Multiplying by a Form of $1$

Evaluate

$$ \int \sec x ,,dx $$

Similarly, one can find

$$ \int \csc x,, dx $$

§ 8.2 Integration by Parts

Integration by parts is a technique for simplifying integrals of the form

$$ \int f(x)g(x)dx. $$

It is useful when $f$ can be differentiated repeatedly and $g$ can be integrated repeatedly without difficulty. For example the integral $\int x e^x dx$ and $\int e^x \sin(x)d x$ can be such type.

Product Rule in Integral Form

Integration by Parts Formula

$$ \int u(x), v'(x) , dx = u(x), v(x) -\int u'(x) , v(x) , dx. $$

The mindset behind the formula is that to transfer a difficult integration to an easier one.

Sometimes it is easier to remember the formula if we write it in differential form.

$$ \int u dv = uv - \int v du. $$

Why it works?

The formula for integration by parts is coming from the Product Rule for derivatives

$$ \frac{d}{dx}\left(u(x) v(x)\right) = u'(x) v(x) + u(x) v'(x). $$

Example 1. Using Integration by Parts

Find

$$ \int x\cos(x) , dx. $$

Example 3. Integral of the Natural Logarithm

Find

$$ \int \ln(x) dx. $$

Example 4. Repeated Use of Integration by Parts

Find

$$ \int x^2 e^x dx $$

Example 5. Solving for the Unknown Integral (An Circling-Back Integration)

Find

$$ \int e^x \cos(x) dx. $$

Evaluating Definite Integrals by Parts

Combining the Fundamental Theorem of Calculus with the Product Rule, we find the following formula for definite integrals

$$ u(x) v(x) \bigg]_a^b = \int_a^b u'(x) v(x) dx + \int_a^b u(x) v'(x) dx. $$

Therefore we can find the following integration by parts formula for definite integrals.

Integration by Parts Formula for Definite Integrals

$$ \int_a^b u(x) v'(x) dx= u(x) v(x) \bigg]_a^b - \int_a^b u'(x) v(x) dx. $$

Example 6. Finding Area

Find the area of the region bounded by the curve $y=x e^{-x}$ and the $x$-axis from $x=0$ to $x=4.$

Example 7. Repeating Multiple Times. Ft. The Tabular Integration Method

Find

$$ \int x^3 \sin(x) dx. $$

Example 9. A Reduction Formula (Iterative Formulas)

Find the integral

$$ I_n = \int \cos^n(x) dx $$

Example 10. Using the Reduction Formula

Find

$$ \int \cos^3(x) dx. $$