Podcast
Questions and Answers
What is the result of the integral ∫ tan x dx?
What is the result of the integral ∫ tan x dx?
- log tan x + C (correct)
- log sec x + C (correct)
- -log cos x + C
- sin x + C
What substitution is used for the integral ∫ sec x dx?
What substitution is used for the integral ∫ sec x dx?
- cosec x + cot x = t
- tan x + sec x = t (correct)
- sin x = t
- cos x = t
Which of the following integrals results in log sin x + C?
Which of the following integrals results in log sin x + C?
- ∫ cot x dx (correct)
- ∫ tan x dx (correct)
- ∫ cosec x dx
- ∫ sec x dx
What is the result of the integral ∫ cosec x dx?
What is the result of the integral ∫ cosec x dx?
When finding the integral ∫ sin x cos² x dx, which simplification is applied?
When finding the integral ∫ sin x cos² x dx, which simplification is applied?
What is the final form of the integral ∫ 1 / (cosec² x - cot² x) dx?
What is the final form of the integral ∫ 1 / (cosec² x - cot² x) dx?
What integral corresponds to the substitution cos x = t?
What integral corresponds to the substitution cos x = t?
What is the relationship established in ∫ (1 + tan x) dx?
What is the relationship established in ∫ (1 + tan x) dx?
What technique is primarily used to solve the integrals involving trigonometric functions?
What technique is primarily used to solve the integrals involving trigonometric functions?
What is the integral of $x^2 + 2x + 2$?
What is the integral of $x^2 + 2x + 2$?
Which of the following correctly describes a rational function?
Which of the following correctly describes a rational function?
What characterizes a proper rational function?
What characterizes a proper rational function?
If $P(x)$ is of degree 2 and $Q(x)$ is of degree 3, what type of rational function is it?
If $P(x)$ is of degree 2 and $Q(x)$ is of degree 3, what type of rational function is it?
Which of these expressions represents $(x - 1)(x - 2)$ expanded?
Which of these expressions represents $(x - 1)(x - 2)$ expanded?
Which polynomial is the result of combining $7 - 6x - x^2$ in standard form?
Which polynomial is the result of combining $7 - 6x - x^2$ in standard form?
What does the expression $\frac{4x - x^2}{x + 2}$ simplify to when factored correctly?
What does the expression $\frac{4x - x^2}{x + 2}$ simplify to when factored correctly?
What is the result of the integral $\int (x^3 + 1) dx$?
What is the result of the integral $\int (x^3 + 1) dx$?
Which integral represents the anti-derivative of $4x^3 - 6$?
Which integral represents the anti-derivative of $4x^3 - 6$?
What is the result of $\int (sin x + cos x) dx$?
What is the result of $\int (sin x + cos x) dx$?
What is the value of $\int cosec x (cosec x + cot x) dx$?
What is the value of $\int cosec x (cosec x + cot x) dx$?
What is the integral of $\int cos^2 x dx$ equivalent to?
What is the integral of $\int cos^2 x dx$ equivalent to?
What does the constant $C$ represent in the context of integration?
What does the constant $C$ represent in the context of integration?
Which of the following statements about the integral $\int x^n dx$ for $n = 3$ is true?
Which of the following statements about the integral $\int x^n dx$ for $n = 3$ is true?
In the integration process, what does splitting integrals help to achieve?
In the integration process, what does splitting integrals help to achieve?
What is the incorrect result of $\int (x^2 + 2 e^x) dx$?
What is the incorrect result of $\int (x^2 + 2 e^x) dx$?
What is the derived function of $x^4 - 6x$?
What is the derived function of $x^4 - 6x$?
What is the meaning of the symbol ∫ f (x) dx?
What is the meaning of the symbol ∫ f (x) dx?
In the integral ∫ f (x) dx, what does f (x) represent?
In the integral ∫ f (x) dx, what does f (x) represent?
Which formula correctly represents the integral of x with respect to x?
Which formula correctly represents the integral of x with respect to x?
What does the constant of integration represent in an integral?
What does the constant of integration represent in an integral?
What is the integral of cos x with respect to x?
What is the integral of cos x with respect to x?
Which of the following formulas represents the integral of sec² x?
Which of the following formulas represents the integral of sec² x?
What is the integral of e^x with respect to x?
What is the integral of e^x with respect to x?
According to the standard formulae, what is the integral of -cos x?
According to the standard formulae, what is the integral of -cos x?
In the context of integration, what does Integration refer to?
In the context of integration, what does Integration refer to?
What is the integral of 1/x with respect to x?
What is the integral of 1/x with respect to x?
Which formula represents the integral of cosec² x?
Which formula represents the integral of cosec² x?
What is the result of integrating sec x tan x with respect to x?
What is the result of integrating sec x tan x with respect to x?
Which integral represents the antiderivative of the function e^x?
Which integral represents the antiderivative of the function e^x?
What does the integral ∫ sec² x dx equal?
What does the integral ∫ sec² x dx equal?
What does the formula for the integral of the product of two functions state?
What does the formula for the integral of the product of two functions state?
What choice of functions would make integration by parts applicable for $,\int x ,cos ,x ,dx$?
What choice of functions would make integration by parts applicable for $,\int x ,cos ,x ,dx$?
What is the result of the integral $,\int x ,cos ,x ,dx$ when using the correct choice of functions?
What is the result of the integral $,\int x ,cos ,x ,dx$ when using the correct choice of functions?
Why is integration by parts not applicable for the integral $\int x ,sin ,x ,dx$?
Why is integration by parts not applicable for the integral $\int x ,sin ,x ,dx$?
What integral is established when applying integration by parts to $,\int x ,cos ,x ,dx$?
What integral is established when applying integration by parts to $,\int x ,cos ,x ,dx$?
What is the form of the equation derived for $,\int f(x) g(x) ,dx$?
What is the form of the equation derived for $,\int f(x) g(x) ,dx$?
What is the first step in applying integration by parts to the integral $,\int x ,cos ,x ,dx$?
What is the first step in applying integration by parts to the integral $,\int x ,cos ,x ,dx$?
Which expression correctly represents integration by parts for $,\int x ,cos ,x ,dx$?
Which expression correctly represents integration by parts for $,\int x ,cos ,x ,dx$?
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Study Notes
Integration
- Integral of f with respect to x: ∫ f (x) dx
- Integrand: f (x) in ∫ f (x) dx
- Variable of integration: x in ∫ f (x) dx
- Integrate: find the integral
- An integral of f: a function F such that F′(x) = f (x)
- Integration: the process of finding the integral
- Constant of Integration: any real number C, considered as a constant function
Standard integration formulas
- ∫ x^n dx = (x^(n+1))/(n+1) + C, n ≠ –1
- ∫ dx = x + C
- ∫ cos x dx = sin x + C
- ∫ sin x dx = – cos x + C
- ∫ sec^2 x dx = tan x + C
- ∫ cosec^2 x dx = – cot x + C
- ∫ sec x tan x dx = sec x + C
- ∫ cosec x cot x dx = – cosec x + C
- ∫ dx/(1 – x^2) = sin^-1 x + C
- ∫ dx/(1 – x^2) = – cos^-1 x + C
- ∫ dx/(1 + x^2) = tan^-1 x+C
- ∫ e^x dx = e^x +C
- ∫ dx/x = log | x | +C
- ∫ a^x dx = a^x/log a +C
Integration by parts
- Used for integrating products of two functions
- Formula: ∫ f (x) g (x) dx = f (x) ∫ g (x) dx – ∫ [ ∫ g (x) dx] f ′(x ) dx
- "The integral of the product of two functions = (first function) × (integral of the second function) – Integral of [(differential coefficient of the first function) × (integral of the second function)]"
- The choice of the first and second function is important for the simplification of the integral
- Integration by parts is not applicable to all products of functions. For example, for ∫ x sin x dx, there is no function whose derivative is x sin x.
- While finding the integral of the second function, no constant of integration is added.
Standard integrals of trigonometric functions
- ∫ tan x dx = log sec x + C
- ∫ cot x dx = log sin x + C
- ∫ sec x dx = log (sec x + tan x) + C
- ∫ cosec x dx = log (cosec x – cot x) + C
Techniques for complex integration
- Use substitution to simplify the integrand
- Split the integrand into simpler terms
- Use integration by parts to simplify complex products of functions.
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