Techniques of Integration flashcards

$\frac{x^4}{4}+c$x^4/4 + c

from “The Antiderivative”

$\frac{x^2}{2}+c$x^2/2 + c

from “The Antiderivative”

$8x^5+c$8x^5 + c

from “Integrating Powers”

$4x^5+c$4x^5 + c

from “Integrating Powers”

3

from “Finding the Constant of Integration”

2

from “Finding the Constant of Integration”

"the integral from 0 to 3 of x squared, dx"

from “Reading the Integral Sign”

where the sweep starts and stops

from “Reading the Integral Sign”

$3x^{2}dx$3x² dx

from “Differentials”

2x dx

from “Differentials”

an exact area

from “The Definite Integral”

the limit those totals approach

from “The Definite Integral”

the left edge of the strip

from “Left, Right and Midpoint Riemann Sums”

7.5

from “Left, Right and Midpoint Riemann Sums”

the definite integral

from “The Riemann Sum in Sigma Notation”

$a+i\Delta x$a + i Δx

from “The Riemann Sum in Sigma Notation”

16

from “The Fundamental Theorem of Calculus”

25

from “The Fundamental Theorem of Calculus”

6

from “Area Below the Axis”

negative

from “Area Below the Axis”

−7

from “Properties of the Definite Integral”

$\int fdx+\int gdx$∫ f dx + ∫ g dx

from “Properties of the Definite Integral”

$x^2+1$x² + 1

from “Integration by Substitution”

$\frac{{\left(x^2+1\right)}^6}{6}+c$(x² + 1)⁶/6 + c

from “Integration by Substitution”

$\frac{e^{5x}}{5}+c$e^(5x)/5 + c

from “Integrating f(ax + b)”

$\sin \frac{2x}{2}+c$sin 2x/2 + c

from “Integrating f(ax + b)”

x

from “Integration by Parts”

$uv-\int vdu$uv − ∫v du

from “Integration by Parts”

sin x + c

from “Integrating Trigonometric Functions”

−cos x + c

from “Integrating Trigonometric Functions”

$\frac{\left(1-\cos 2x\right)}{2}$(1 − cos 2x)/2

from “Integrating sin² and cos²”

the double angle formula for cos 2A

from “Integrating sin² and cos²”

$e^x+c$eˣ + c

from “Integrating Exponentials and Logarithms”

ln|x| + c

from “Integrating Exponentials and Logarithms”

$\frac{A}{\left(x-1\right)}+\frac{B}{{\left(x-1\right)}^2}$A/(x−1) + B/(x−1)²

from “Integration by Partial Fractions”

Ax + B

from “Integration by Partial Fractions”

$\ln |x^3+5|+c$ln|x³ + 5| + c

from “Integrals of f′ over f”

the derivative of the denominator

from “Integrals of f′ over f”

−ln|cos x| + c

from “Integrating sec²x, sec x tan x and tan x”

ln|sin x| + c

from “Integrating sec²x, sec x tan x and tan x”

$\left(\frac{1}{2}\right){\tan }^{-1}\left(\frac{x}{2}\right)+c$(1/2) tan⁻¹(x/2) + c

from “Integrals That Give tan⁻¹ and sin⁻¹”

${\sin }^{-1}x+c$sin⁻¹x + c

from “Integrals That Give tan⁻¹ and sin⁻¹”

$1+{\tan }^2\theta ={\sec }^2\theta$1 + tan²θ = sec²θ

from “Choosing a Trigonometric Substitution”

$x=5\sin \theta$x = 5 sin θ

from “Choosing a Trigonometric Substitution”

arsinh is a logarithm already

from “Integrals That Give arsinh and arcosh”

arsinh x + c

from “Integrals That Give arsinh and arcosh”