Podcast
Questions and Answers
What is the integral of $\frac{1}{x}$ with respect to $x$?
What is the integral of $\frac{1}{x}$ with respect to $x$?
- $ln|x| + C$ (correct)
- $\frac{1}{x^2} + C$
- $\frac{1}{x^2} + C$
- $-ln|x| + C$
What is the result of the integral $\int \frac{1}{4x - 1} dx$?
What is the result of the integral $\int \frac{1}{4x - 1} dx$?
- $\frac{1}{4} ln|4x - 1| + C$ (correct)
- $4 ln|4x - 1| + C$
- $ln|4x - 1| + C$
- $ln|x - \frac{1}{4}| + C$
If $u = 4x - 1$, then what is $du$?
If $u = 4x - 1$, then what is $du$?
- $4 dx$ (correct)
- $4x dx$
- $dx$
- $-4 dx$
What type of function does $\int \frac{x}{x^2 + 1} dx$ involve?
What type of function does $\int \frac{x}{x^2 + 1} dx$ involve?
Which of the following is a common method used when integrating transcendental functions?
Which of the following is a common method used when integrating transcendental functions?
Which type of function is $f(x) = ln(x)$?
Which type of function is $f(x) = ln(x)$?
What is the integral of $e^x$ with respect to $x$?
What is the integral of $e^x$ with respect to $x$?
Which of the following is an example of an inverse trigonometric function?
Which of the following is an example of an inverse trigonometric function?
What is the integral of $\frac{du}{\sqrt{a^2 - u^2}}$?
What is the integral of $\frac{du}{\sqrt{a^2 - u^2}}$?
What is the integral of $\frac{du}{a^2 + u^2}$?
What is the integral of $\frac{du}{a^2 + u^2}$?
What is the relationship between $sec(x)$ and $tan(x)$ in the context of trigonometric identities?
What is the relationship between $sec(x)$ and $tan(x)$ in the context of trigonometric identities?
What is the result of the integral $\int \frac{sec(x)tan(x)}{sec(x) - 1} dx$?
What is the result of the integral $\int \frac{sec(x)tan(x)}{sec(x) - 1} dx$?
Given the equation $3x - x^2 = -[x^2 - 3x + (\frac{3}{2})^2 - (\frac{3}{2})^2]$, what principle allows us to add and subtract $(\frac{3}{2})^2$ inside the brackets?
Given the equation $3x - x^2 = -[x^2 - 3x + (\frac{3}{2})^2 - (\frac{3}{2})^2]$, what principle allows us to add and subtract $(\frac{3}{2})^2$ inside the brackets?
If $\int f(x) dx = F(x) + C$, what does 'C' represent?
If $\int f(x) dx = F(x) + C$, what does 'C' represent?
Integration of which function results into logarithmic function?
Integration of which function results into logarithmic function?
Which of these functions is classified as a transcendental function?
Which of these functions is classified as a transcendental function?
What is a general strategy for integrating functions involving $e^x$ when u-substitution is applicable?
What is a general strategy for integrating functions involving $e^x$ when u-substitution is applicable?
In the context of integration, what is the purpose of 'multiplying and dividing by 4' when evaluating $\int \frac{1}{4x - 1} dx$?
In the context of integration, what is the purpose of 'multiplying and dividing by 4' when evaluating $\int \frac{1}{4x - 1} dx$?
Which identity is useful when evaluating the definite integral of $sin^2(x)$ or $cos^2(x)$?
Which identity is useful when evaluating the definite integral of $sin^2(x)$ or $cos^2(x)$?
Flashcards
Transcendental Functions
Transcendental Functions
Functions that are not algebraic, including logarithmic, exponential, trigonometric, inverse trigonometric, and hyperbolic functions.
∫(1/u) du
∫(1/u) du
The integral of 1/u with respect to u is the natural logarithm of the absolute value of u, plus a constant of integration.
∫ du / √(a² - u²)
∫ du / √(a² - u²)
This formula is arcsin(u/a) + C, where 'u' is a function of x, 'a' is a constant, and 'C' is the integration constant.
∫ du / (a² + u²)
∫ du / (a² + u²)
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∫ du / (u√(u² - a²))
∫ du / (u√(u² - a²))
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Study Notes
- Transcendental functions can be integrated
- These include:
- Logarithmic Functions
- Exponential Functions
- Trigonometric Functions
- Inverse Trigonometric Functions
- Hyperbolic Functions
Logarithmic Functions
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To solve ∫ (1 / (4x - 1)) dx
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If u = 4x - 1, then du = 4 dx
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Multiply and divide by 4 to get 1/4 ∫ (1 / (4x - 1)) 4 dx
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Substitute to acheive 1/4 ∫ (1 / u) du, where u = 4x - 1
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Apply Log Rule to get 1/4 ln|u| + C
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Back-substitute 1/4 ln|4x - 1| + C
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The area of the region bounded by the graph of y, the x-axis, and x = 3 for ∫₀³ x / (x² + 1) dx is ½ ln 10
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∫ (e^x - e^-x) / (e^x + e^-x) dx = ln (e^x + e^-x) + C
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∫ sec x tan x / (sec x - 1) dx = ln |sec x - 1| + C
Integrals Involving Inverse Trigonometric Functions
- Theorem 5.17 describes integrals with inverse trig functions
- Let u be a differentiable function of x, and a > 0.
- ∫ du / √(a² - u²) = arcsin(u/a) + C
- ∫ du / (a² + u²) = 1/a arctan(u/a) + C
- ∫ du / (u√(u² - a²)) = 1/a arcsec(|u|/a) + C
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