Integral as an antiderivative - One Line Questions
1.
Find the antiderivative of $f(x) = \sin(2x)$. —
$-\frac{1}{2}\cos(2x) + C$
2.
Find the antiderivative of $f(x) = \frac{1}{x^3}$. —
$-\frac{1}{2x^2} + C$
3.
What is the antiderivative of $f(x) = \frac{1}{x^2}$? —
$-\frac{1}{x} + C$
4.
The antiderivative of $f(x) = \cos(x)$ is: —
$\sin(x) + C$
5.
What is the antiderivative of $f(x) = \frac{x}{\sqrt{1-x^2}}$? —
$-\sqrt{1-x^2} + C$
6.
The antiderivative of $f(x) = \frac{1}{\sqrt{1-x^2}}$ is also known as: —
$-\arccos(x) + C$
7.
The antiderivative of $f(x) = \frac{1}{\sqrt{a^2-x^2}}$ where $a > 0$ is: —
$\arcsin(\frac{x}{a}) + C$
8.
What is the antiderivative of $f(x) = \frac{1}{\sqrt{1-x^2}}$? —
$\arcsin(x) + C$
9.
What is the antiderivative of $f(x) = \frac{1}{1+x^2}$? —
$\arctan(x) + C$
10.
What is the antiderivative of $f(x) = \sin(x)$? —
$-\cos(x) + C$
11.
Find the antiderivative of $f(x) = \sinh(x)$. —
$\cosh(x) + C$
12.
What is the antiderivative of $f(x) = \frac{1}{x^2-a^2}$? —
$\frac{1}{2a}\ln|\frac{x-a}{x+a}| + C$
13.
The antiderivative of $f(x) = e^{3x}$ is: —
$\frac{1}{3}e^{3x} + C$
14.
Find the antiderivative of $f(x) = \frac{1}{x^2+a^2}$ where $a \neq 0$. —
$\frac{1}{a}\arctan(\frac{x}{a}) + C$
15.
If $\int f(x) dx = F(x) + C$, then $\int f(ax+b) dx$ is: —
$\frac{1}{a}F(ax+b) + C$
16.
Find the antiderivative of $f(x) = \sqrt{x}$. —
$\frac{2}{3} x^{3/2} + C$
17.
The antiderivative of $f(x) = a^x$ (where $a > 0, a \neq 1$) is: —
$\frac{a^x}{\ln(a)} + C$
18.
What is the antiderivative of $x^n$, where $n \neq -1$? —
$\frac{x^{n+1}}{n+1}$
19.
Find the antiderivative of $f(x) = x^2$. —
$\frac{x^3}{3} + C$
20.
What is the property of linearity for indefinite integrals? —
Both A and B
21.
The antiderivative of $f(x) = \frac{1}{x}$ for $x < 0$ is: —
$\ln|x| + C$
22.
The antiderivative of $f(x) = \frac{2x}{1+x^2}$ is: —
$\ln|1+x^2| + C$
23.
Find the antiderivative of $f(x) = \frac{1}{x}$ for $x > 0$. —
$\ln(x) + C$
24.
What is the antiderivative of $f(x) = \csc(x)$? —
$\ln|\csc(x) - \cot(x)| + C$
25.
Find the antiderivative of $f(x) = \frac{1}{x \ln(x)}$. —
$\ln|\ln(x)| + C$
26.
The antiderivative of $f(x) = \sec(x)$ is: —
$\ln|\sec(x) + \tan(x)| + C$
27.
What is the antiderivative of $f(x) = \tan(x)$? —
$\ln|\sec(x)| + C$
28.
Find the antiderivative of $f(x) = \cot(x)$. —
$\ln|\sin(x)| + C$
29.
What is the antiderivative of $f(x) = x^{-1}$? —
$\ln|x| + C$
30.
The antiderivative of $f(x) = x^n$ for $n=-1$ is: —
$\ln|x| + C$
31.
Consider the function $f(x) = \frac{d}{dx}(\sin(x^2))$. What is the integral of $f(x)$? —
$\sin(x^2) + C$
32.
What is the antiderivative of $f(x) = \cosh(x)$? —
$\sinh(x) + C$
33.
The antiderivative of $f(x) = \sec^2(x)$ is: —
$\tan(x) + C$
34.
What is the antiderivative of $f(x) = \frac{1}{x \sqrt{x^2-1}}$? —
$\text{arcsec}(|x|) + C$
35.
Find the antiderivative of $f(x) = 3x^4 - 2x + 5$. —
$\frac{3x^5}{5} - x^2 + 5x + C$
36.
The antiderivative of $f(x) = \frac{1}{\sqrt{x}}$ is: —
$2\sqrt{x} + C$
37.
What is the antiderivative of $f(x) = x^3$? —
$\frac{x^4}{4} + C$
38.
What is the antiderivative of $f(x) = c$, where c is a constant? —
$cx + C$
39.
What is the antiderivative of $f(x) = e^x$? —
$e^x + C$
40.
If $F'(x) = f(x)$, then the indefinite integral of $f(x)$ is: —
$F(x) + C$
41.
If $\int f(x) dx = F(x) + C$, then what is $\int (f(x) + g(x)) dx$? —
$F(x) + G(x) + C$
42.
If $F(x)$ is an antiderivative of $f(x)$, then the antiderivative of $f(x+a)$ is: —
$F(x+a) + C$
43.
If $\int f(x) dx = G(x) + C$, then what is $\int g(x) dx$ if $g(x) = k \cdot f(x)$ for a constant $k$? —
$k \cdot G(x) + C$
44.
What is the antiderivative of $f(x) = \ln(x)$? —
$x \ln(x) - x + C$
45.
The antiderivative of $f(x) = x \cos(x)$ is found using integration by parts. The result is: —
$x \sin(x) + \cos(x) + C$
46.
If $F(x)$ is an antiderivative of $f(x)$, what is $F(x) + C$? —
The general antiderivative
47.
If F(x) is an antiderivative of f(x), what is the relationship between F'(x) and f(x)? —
F'(x) = f(x)
48.
What is the general form of the antiderivative of a function f(x)? —
F(x) + C
49.
What is the fundamental concept represented by an integral as an antiderivative? —
Reversing the process of differentiation
50.
The constant 'C' in the indefinite integral $\int f(x) dx = F(x) + C$ is known as the: —
Constant of integration