Podcast Beta
Questions and Answers
What is the derivative of Sec x?
Which formula represents the product rule for derivatives?
What is the derivative of ln(x)?
What is the correct formula for the quotient rule?
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What is the derivative of e^x?
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Which of the following statements correctly describes the function $f(x) = 3x$?
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What is the result of the integration $ rac{1}{a} Sec^2(ax + b)dx$?
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What is the integral of $cos(ax + b) dx$?
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In the context of integration, which property is exemplified by $ rac{1}{a} rac {(ax + b)^{n+1}}{n+1} + c$?
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Which of the following represents the correct transformation for $ an(x + a)$?
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What is the integral of $ rac{1}{2x + 3}$ with respect to $x$?
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Which formula correctly represents the integral of $e^{3x + 2}$?
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What is the result of integrating $4x + 5$?
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What is the integral of $ an{x}$?
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Which of the following is the correct formula for the integral of $ rac{1}{ ext{cosec}x} ext{cot}x$?
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What is the integral of $ an x$?
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What is the result of the integration $ rac{1}{ an^2 x} dx$?
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Which of the following is the integral of $ rac{1}{ an x} dx$?
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What is the integral of $ rac{1}{x} dx$?
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What is the integral of $ rac{1}{ an^2 x}$?
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Study Notes
Indefinite Integration
- Derivative of x with respect to x is nx; integral of x dx equals x²/2 + c.
- Derivative of sin x is cos x; integral of cos x dx equals sin x + c.
- Derivative of cos x is -sin x; integral of sin x dx equals -cos x + c.
- Derivative of tan x is sec² x; integral of sec² x dx equals tan x + c.
- Derivative of cosec x is -cosec x cot x; integral of cosec x cot x dx equals -cosec x + c.
- Derivative of sec x is sec x tan x; integral of sec x tan x dx equals sec x + c.
- Derivative of cot x is -cosec² x; integral of cosec² x dx equals -cot x + c.
- Exponential derivative, d/dx(e^x) = e^x; integral, ∫e^x dx equals e^x + c.
- Derivative of ln(x) is 1/x; integral, ∫(a^x) dx = (a^x/ln a) + c.
- Integral of 1/√(1-x²) dx is sin⁻¹(x) + c; integral of -1/√(1-x²) dx is cos⁻¹(x) + c.
Relations and Functions
- Function f(x) = x behaves as follows:
- Positive for x > 0
- Zero for x = 0
- Negative for x < 0
- Assume f(x) = -1 if x < 0.
- Analyze injectivity, surjectivity, and bijection for functions:
- f(x) = x is bijective (both injective and onto).
- f(x) = 3x is also bijective.
Integrations
- Integral of a constant multiplied by dx equals x + c.
- Integral of the sum of two functions is the sum of their integrals.
- Integral of k times f(x) dx is k times the integral of f(x) dx.
Substitution Method
- Formula for ∫(ax + b)ⁿ dx: (1/a) * (ax + b)^(n+1)/(n+1) + c.
- ∫sec²(ax + b)dx = (1/a)tan(ax + b) + c.
- ∫sin(ax + b) dx = -(1/a)cos(ax + b) + c.
- ∫cos(ax + b) dx = (1/a)sin(ax + b) + c.
- ∫cosec²(ax + b)dx = -(1/a)cot(ax + b) + c.
- ∫sec(ax + b)tan(ax + b) dx = (1/a)sec(ax + b) + c.
- ∫cosec(ax + b)cot(ax + b)dx = -(1/a)cosec(ax + b) + c.
Integration Formulas
- ∫(1/(ax+b)) dx = (1/a)ln|ax+b| + c.
- ∫e^(ax+b) dx = (1/a)e^(ax+b) + c.
- ∫(ax+b) dx = (a/2)x² + bx + c.
- ∫tan x dx = ln|sec x| + c.
- ∫cot x dx = ln|sin x| + c.
- ∫cosec x dx = ln|cosec x - cot x| + c.
- ∫sec x dx = ln|sec x + tan x| + c.
- For f'(x)f(x) integral, use (1/a)ln|f(x)| + c for linear functions.
Methods of Differentiation
- Derivative of sec x is sec x tan x.
- Derivative of cot x is -cosec² x.
- Derivative of tan x is sec² x.
- Derivative of cosec x is -cosec x cot x.
- Derivative of ln x is 1/x.
- Derivative of e^x is e^x.
- Derivative of a^x is a^x ln a.
- Derivative of arccot x is -1/(1 + x²).
- Derivative of arcsec x is 1/(|x|√(x² - 1)).
- Derivative of arccosec x is -1/(|x|√(x² - 1)).
Properties of Derivatives
- The derivative of a sum d/dx[f(x) + g(x)] = d/dx f(x) + d/dx g(x).
- Scaling property: d/dx[kf(x)] = k d/dx f(x).
- Product rule: d/dx(uv) = u(d/dx v) + v(d/dx u).
- Quotient rule: d/dx(u/v) = (v(d/dx u) - u(d/dx v))/v².
- Chain rule: d/dx[f(g(x))] = f'(g(x))(d/dx g(x)).
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Description
This quiz covers the fundamental concepts of indefinite integration in integral calculus. You will encounter various integration rules and their corresponding derivatives. Test your knowledge of integrating functions and learn through practice problems.
