Integration by substitution by parts and by partial fractions - One Line Questions
1.
What is the integral of ∫ tan(x) dx? —
-ln|cos(x)| + C
2.
What is the result of ∫ x*sin(x) dx? —
-x*cos(x) + sin(x) + C
3.
Evaluate ∫ (1 / (x^2 + 4)) dx. —
(1/2) * arctan(x/2) + C
4.
What is the integral of ∫ x / (x^2 + 1) dx? —
(1/2) * ln(x^2 + 1) + C
5.
Evaluate ∫ (1 / (x^2 - 1)) dx using partial fractions. —
(1/2) * ln|(x-1)/(x+1)| + C
6.
What is the integral of ∫ x * cos(x^2) dx using substitution? —
(1/2) * sin(x^2) + C
7.
What is the integral of 1/(x^2 + a^2) dx? —
(1/a) * arctan(x/a) + C
8.
If ∫ f(x) dx = F(x) + C, then ∫ f(ax+b) dx = ? —
(1/a) * F(ax+b) + C
9.
For a repeated irreducible quadratic factor (ax^2 + bx + c)^n, the partial fraction decomposition includes terms of the form: —
(A_1x + B_1)/(ax^2+bx+c) + ... + (A_nx + B_n)/(ax^2+bx+c)^n
10.
An irreducible quadratic factor in the denominator (ax^2 + bx + c) leads to a partial fraction of the form: —
(Ax + B) / (ax^2 + bx + c)
11.
What is the integral of ∫ x * arctan(x) dx? —
(x^2/2)*arctan(x) - (1/2)*x + (1/2)*arctan(x) + C
12.
Using substitution t = x+1, evaluate ∫ (x+2)/(x+1) dx. —
(x+1) + ln|x+1| + C
13.
Which integral is best suited for integration by partial fractions? —
∫ (2x+1) / (x^2-1) dx
14.
What is the standard formula for integration by parts? —
∫ u dv = uv - ∫ v du
15.
Which integral can be solved using integration by parts? —
∫ x*cos(x) dx
16.
Which integral requires integration by parts due to the product of a polynomial and an exponential function? —
∫ x*e^(3x) dx
17.
What is the partial fraction decomposition of 1 / (x(x+1))? —
1/x - 1/(x+1)
18.
Consider the integral ∫ x^2 * e^x dx. How many times would integration by parts need to be applied? —
2
19.
For a repeated linear factor (ax+b)^n in the denominator, the partial fraction decomposition includes terms up to: —
A_n / (ax+b)^n
20.
For the integral ∫ (3x + 5) / ((x-1)(x+2)) dx, the partial fraction decomposition will be of the form: —
A/(x-1) + B/(x+2)
21.
Which rule is fundamental to integration by parts? —
Product Rule
22.
The method of substitution is essentially reversing the: —
Chain Rule
23.
What is the first step in integrating a rational function using partial fractions? —
Decompose the rational function into simpler fractions.
24.
What is the derivative of u in the integration by parts formula ∫ u dv = uv - ∫ v du? —
du
25.
If we choose u = x and dv = e^(-x) dx for ∫ x*e^(-x) dx, what is du and v? —
du = dx, v = -e^(-x)
26.
If ∫ f(x) dx = G(x) + C, and we use substitution x = g(t), then ∫ f(g(t))g'(t) dt is equal to: —
G(g(t)) + C
27.
Consider the integral ∫ x^3 dx. Which integration technique is NOT needed? —
Integration by parts
28.
Which method is most suitable for ∫ e^x * sin(x) dx? —
Integration by parts (applied twice)
29.
In integration by parts, the choice of 'u' and 'dv' is crucial. Which mnemonic is often used to decide which function to choose as 'u'? —
LIATE (Logarithmic, Inverse trigonometric, Algebraic, Trigonometric, Exponential)
30.
Which type of factor in the denominator leads to a partial fraction of the form A/(ax+b)? —
Linear factor.
31.
Using substitution, what is the integral of 2x/(1+x^2) dx? —
ln(1+x^2) + C
32.
What is the integral of ∫ (2x+3) / (x^2+3x+5) dx? —
ln(x^2+3x+5) + C
33.
If the degree of the numerator is greater than or equal to the degree of the denominator in a rational function, what is the first step before partial fraction decomposition? —
Perform polynomial long division.
34.
For the integral ∫ (x^2 + 3x + 2) / (x - 1) dx, what is the first step? —
Perform polynomial long division.
35.
For the integral ∫ (x^2 + 1) / (x^2 - 1) dx, the first step is: —
Polynomial long division.
36.
A rational function is a ratio of two: —
Polynomials.
37.
Integration by partial fractions is primarily used for integrating: —
Rational functions.
38.
What is the primary technique used in integration by substitution? —
Replacing a function with a simpler one using a new variable.
39.
If x = tan(θ), then dx = ? —
sec^2(θ) dθ
40.
Which method is most appropriate for ∫ (x+1) / (x^2+2x+2) dx? —
Substitution (let u = x^2+2x+2)
41.
The formula for integration by parts is derived from: —
The derivative of a product of two functions.
42.
What is the integral of dv in the integration by parts formula ∫ u dv = uv - ∫ v du? —
v
43.
Consider ∫ (x^2) / (x - 1) dx. After polynomial division, the integrand becomes: —
x + 1 + 1/(x-1)
44.
What substitution is useful for integrating functions of the form √(x^2 - a^2)? —
x = a*sec(θ)
45.
What substitution is useful for integrating functions of the form √(a^2 - x^2)? —
x = a*sin(θ)
46.
What substitution is useful for integrating functions of the form √(a^2 + x^2)? —
x = a*tan(θ)
47.
What is the integral of ∫ x^2 dx? —
x^3/3 + C
48.
What is the integral of x*e^x dx using integration by parts? —
x*e^x - e^x + C
49.
What is the integral of ∫ e^x * (x+1) dx? —
x*e^x + C
50.
Evaluate ∫ ln(x) dx using integration by parts. —
x*ln(x) - x + C