MAT 1310 Chapters 6-7 Formula and Method Guide

Vocabulary flashcards covering core derivative rules, inverse trig, exponential and logarithmic functions, integration techniques, trig integrals, trig substitution, integration by parts, and partial fractions.

Cotangent-Cosecant Pythagorean Identity

$$1 + \cot^2(x) = \csc^2(x)$$

Product Rule

$$\frac{d}{dx}[f(x)g(x)] = f'(x)g(x) + f(x)g'(x)$$

Quotient Rule

$$\frac{d}{dx}\left[\frac{f(x)}{g(x)}\right] = \frac{g(x)f'(x) - f(x)g'(x)}{(g(x))^2}$$

Chain Rule

If $y = f(g(x))$, then $y' = f'(g(x)) \cdot g'(x)$

Co-function Derivative Sign Rule

Every co-function derivative carries a minus sign: $\cos(u)$, $\cot(u)$, and $\csc(u)$

Variable-Upper-Bound Derivative (FTC Chain Form)

$$\frac{d}{dx}\left[\int_{a}^{g(x)} f(t)\,dt\right] = f(g(x)) \cdot g'(x)$$

Principal Range of $\arcsin(x)$

$\left[-\frac{\pi}{2}, \frac{\pi}{2}\right]$ (Quadrants I and IV)

Principal Range of $\arccos(x)$

$[0, \pi]$ (Quadrants I and II)

Principal Range of $\arctan(x)$

$\left(-\frac{\pi}{2}, \frac{\pi}{2}\right)$ (Quadrants I and IV)

Derivative of $\arcsin(u)$

$$\frac{d}{dx}[\arcsin(u)] = \frac{u'}{\sqrt{1-u^2}}$$

Derivative of $\arctan(u)$

$$\frac{d}{dx}[\arctan(u)] = \frac{u'}{1+u^2}$$

Derivative of \arcsec(u)

\frac{d}{dx}[\arcsec(u)] = \frac{u'}{|u|\sqrt{u^2-1}}

Derivative of $b^u$ (General Base Exponential)

$$\frac{d}{dx}[b^u] = b^u \ln(b) \cdot u'$$

Derivative of $\log_b(u)$ (General Base Logarithm)

$$\frac{d}{dx}[\log_b(u)] = \frac{u'}{u \ln(b)}$$

Integral of $\frac{1}{u}$

$$\int \frac{1}{u}\,du = \ln|u| + C$$

Inverse Sine Integral Form

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

Inverse Tangent Integral Form

$$\int \frac{1}{a^2+u^2}\,du = \frac{1}{a}\arctan\left(\frac{u}{a}\right) + C$$

Integral of $\tan(u)$

$$\int \tan(u)\,du = \ln|\sec(u)| + C = -\ln|\cos(u)| + C$$

Integral of $\sec(u)$

$$\int \sec(u)\,du = \ln|\sec(u) + \tan(u)| + C$$

Trig Integral Decision Rule: $\sin^m(x)\cos^n(x)$ with $n$ Odd

Save one $\cos(x)$; convert remaining using $\cos^2(x) = 1 - \sin^2(x)$; substitute $u = \sin(x)$

Trig Integral Decision Rule: $\sin^m(x)\cos^n(x)$ with $m$ Odd

Save one $\sin(x)$; convert remaining using $\sin^2(x) = 1 - \cos^2(x)$; substitute $u = \cos(x)$

Trig Integral Decision Rule: $\tan^m(x)\sec^n(x)$ with $n$ Even

Save $\sec^2(x)$; convert remaining using $\sec^2(x) = 1 + \tan^2(x)$; substitute $u = \tan(x)$

Trig Integral Decision Rule: $\tan^m(x)\sec^n(x)$ with $m$ Odd

Save $\sec(x)\tan(x)$; convert remaining using $\tan^2(x) = \sec^2(x) - 1$; substitute $u = \sec(x)$

Trigonometric Substitution for $\sqrt{a^2-x^2}$

Substitute $x = a \sin(\theta)$, $dx = a \cos(\theta)\,d\theta$, and use identity $1 - \sin^2(\theta) = \cos^2(\theta)$

Trigonometric Substitution for $\sqrt{a^2+x^2}$

Substitute $x = a \tan(\theta)$, $dx = a \sec^2(\theta)\,d\theta$, and use identity $1 + \tan^2(\theta) = \sec^2(\theta)$

Trigonometric Substitution for $\sqrt{x^2-a^2}$

Substitute $x = a \sec(\theta)$, $dx = a \sec(\theta)\tan(\theta)\,d\theta$, and use identity $\sec^2(\theta) - 1 = \tan^2(\theta)$

Integration by Parts Formula

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

LIATE Priority Rule

Priority order for choosing $u$ in integration by parts: L = logarithmic, I = inverse trig, A = algebraic, T = trig, E = exponential

Partial Fractions Gate 1 Rule

The rational function must be proper (degree of numerator < degree of denominator); if not proper, perform polynomial long division first.

Partial Fraction Decomposition: Distinct Linear Factors

For denominator $(x-a)(x-b)$, required terms are $\frac{A}{x-a} + \frac{B}{x-b}$

Partial Fraction Decomposition: Repeated Linear Factor

For denominator $(x-a)^m$, required terms are $\frac{A_1}{x-a} + \frac{A_2}{(x-a)^2} + \dots + \frac{A_m}{(x-a)^m}$

Partial Fraction Decomposition: Irreducible Quadratic Factor

For denominator $x^2+bx+c$, required term is $\frac{Ax+B}{x^2+bx+c}$