Podcast
Questions and Answers
What is the result of integrating the function $|x-1|$ from 1 to 2?
What is the result of integrating the function $|x-1|$ from 1 to 2?
- 0
- 1 (correct)
- 2
- 3
When evaluating the integral $, ∫_0^2 (4x^3 - 5x^2 + 6x + 9) dx$, which of the following is true about the result?
When evaluating the integral $, ∫_0^2 (4x^3 - 5x^2 + 6x + 9) dx$, which of the following is true about the result?
- The result is a negative number
- The result is an integer (correct)
- The result is equal to 3
- The result is affected by the properties of discontinuity
Which property of definite integrals states that changing the limits of integration negates the value of the integral?
Which property of definite integrals states that changing the limits of integration negates the value of the integral?
- Reversal property (correct)
- Symmetry property
- Linearity property
- Substitution property
What is the integral $, ∫_0^1 x/(x^2+1) dx$ evaluated to?
What is the integral $, ∫_0^1 x/(x^2+1) dx$ evaluated to?
What is the area under the curve for $\int_0^π/4 tan² x dx$?
What is the area under the curve for $\int_0^π/4 tan² x dx$?
What is the first step in using integration by partial fractions?
What is the first step in using integration by partial fractions?
In integration by parts, which function should be chosen as 'u' when applying the ILATE rule?
In integration by parts, which function should be chosen as 'u' when applying the ILATE rule?
If the degree of P(x) is greater than the degree of Q(x) in an integral of the form ∫P(x)/Q(x) dx, what should be done first?
If the degree of P(x) is greater than the degree of Q(x) in an integral of the form ∫P(x)/Q(x) dx, what should be done first?
Which of the following types of integrals can benefit from integration by partial fractions?
Which of the following types of integrals can benefit from integration by partial fractions?
Which set of functions is correctly aligned with the ILATE rule for integration by parts?
Which set of functions is correctly aligned with the ILATE rule for integration by parts?
Flashcards
Integration by Partial Fractions
Integration by Partial Fractions
A method used to integrate rational functions (polynomials divided by polynomials) by decomposing the function into simpler fractions.
Partial Fraction Decomposition
Partial Fraction Decomposition
The process of expressing a complex rational function as a sum of simpler rational functions.
Rational Function
Rational Function
A function that can be expressed as a ratio of two polynomials.
∫(px+q)/((x+a)(x+b)) dx
∫(px+q)/((x+a)(x+b)) dx
Signup and view all the flashcards
∫uv dx
∫uv dx
Signup and view all the flashcards
ILATE rule
ILATE rule
Signup and view all the flashcards
Integration by Parts
Integration by Parts
Signup and view all the flashcards
Definite Integral Property (i)
Definite Integral Property (i)
Signup and view all the flashcards
Definite Integral Property (ii)
Definite Integral Property (ii)
Signup and view all the flashcards
∫_a^b f(x) dx = -∫_b^a f(x) dx
∫_a^b f(x) dx = -∫_b^a f(x) dx
Signup and view all the flashcards
∫_0^π/2 cos²x dx
∫_0^π/2 cos²x dx
Signup and view all the flashcards
∫_1^2 (4x³-5x²+6x+9) dx
∫_1^2 (4x³-5x²+6x+9) dx
Signup and view all the flashcards
∫_0^π/2 √(sin ф cos ф) cos ф dф
∫_0^π/2 √(sin ф cos ф) cos ф dф
Signup and view all the flashcards
∫_{π/6}^{π/3} √(3sin x + cos x)/√(sin 2x) dx
∫_{π/6}^{π/3} √(3sin x + cos x)/√(sin 2x) dx
Signup and view all the flashcards
∫_1^2 |x-1| dx
∫_1^2 |x-1| dx
Signup and view all the flashcards
∫_0^π/4 log (1+tan x) dx
∫_0^π/4 log (1+tan x) dx
Signup and view all the flashcards
Evaluation of Definite Integral by Substitution
Evaluation of Definite Integral by Substitution
Signup and view all the flashcards
Substitution Method Steps
Substitution Method Steps
Signup and view all the flashcards
Study Notes
Integration and Its Properties
- Integration is the inverse process of differentiation
- Anti-derivatives are functions that could have possibly given a function as a derivative
- Indefinite integral gives all the anti-derivatives of a function
- Definite integral finds a numerical value for an area under a curve
- Integration formulas are used for evaluating indefinite and definite integrals in calculus
Integration as an Inverse Process of Differentiation
- If the derivative of F(x) is equal to f(x), then F(x) is called the integral of f(x)
- F(x) + C is also an anti-derivative of f(x), where C is an arbitrary constant (constant of integration)
- There exist infinitely many anti-derivatives of a function
Symbols/Terms/Phrases related to integration
- ∫f(x)dx: Integral of f with respect to x
- f(x): Integrand
- x: Variable of integration
- Integrate: Find the integral
- Integral of f(x): A function F such that F'(x) = f(x)
- Constant of integration: An arbitrary constant
Some Standard Formulae
- ∫xⁿdx = (xⁿ⁺¹)/(n+1) + C, where n ≠ -1
- ∫sin x dx = -cos x + C
- ∫cos x dx = sin x + C
- ∫sec² x dx = tan x + C
- ∫cosec² x dx = -cot x + C
- ∫sec x tan x dx = sec x + C
- ∫cosec x cot x dx = -cosec x + C
- ∫ex dx = ex + C
- ∫1/x dx = ln|x| + C
Studying That Suits You
Use AI to generate personalized quizzes and flashcards to suit your learning preferences.
