Evaluation of definite integrals and properties - One Line Questions
1.
What is the value of \int_{0}^{1} e^{2x} dx? —
(e^2-1)/2
2.
Evaluate \int_{0}^{1} x e^{x^2} dx. —
(e-1)/2
3.
What is the value of \int_{0}^{1} \frac{dx}{4+x^2}? —
\frac{1}{2} \arctan(1/2)
4.
If f(x) is an even function, what is the relation between \int_{-a}^{a} f(x) dx and \int_{0}^{a} f(x) dx? —
\int_{-a}^{a} f(x) dx = 2 \int_{0}^{a} f(x) dx
5.
What is the value of \int_{0}^{\pi} x \sin(x) dx? —
\pi
6.
What is the value of \int_{0}^{\pi} \frac{x}{1+\sin^2(x)} dx? —
\pi^2/2\sqrt{2}
7.
What is the value of \int_{0}^{\pi} \frac{x \sin(x)}{1+\cos^2(x)} dx? —
\pi^2/4
8.
Using the property \int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx, evaluate \int_{0}^{\pi} x \cos^2(x) dx. —
\pi^2/4
9.
What is the value of \int_{0}^{\pi} x \sin^2(x) dx? —
\pi^2/4
10.
Using the property \int_{0}^{2a} f(x) dx = 2 \int_{0}^{a} f(x) dx if f(2a-x) = f(x), evaluate \int_{0}^{\pi} \sin^2(x) dx. —
\pi/2
11.
Evaluate \int_{0}^{1} \frac{dx}{\sqrt{1-x^2}}. —
\pi/2
12.
Using the property \int_{0}^{2a} f(x) dx = 2 \int_{0}^{a} f(x) dx if f(2a-x) = f(x), evaluate \int_{0}^{\pi} \cos^2(x) dx. —
\pi/2
13.
Using the property \int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx, evaluate \int_{0}^{\pi/2} \sin^2(x) dx. —
\pi/4
14.
Evaluate \int_{0}^{\pi/2} \cos^2(x) dx. —
\pi/4
15.
What is the value of \int_{0}^{1} \frac{dx}{1+x^2}? —
\pi/4
16.
Evaluate \int_{0}^{1} \frac{dx}{x^2+1} using the property \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx. —
\pi/4
17.
What is the value of \int_{0}^{\pi/2} \frac{\sin^2(x)}{\sin(x)+\cos(x)} dx? —
\pi/4
18.
Evaluate \int_{0}^{\pi/2} \frac{\sqrt{\sin(x)}}{\sqrt{\sin(x)}+\sqrt{\cos(x)}} dx. —
\pi/4
19.
Evaluate \int_{0}^{\pi} \sin(x) dx. —
2
20.
Calculate \int_{-\pi}^{\pi} \sin^3(x) dx. —
0
21.
What is the value of \int_{0}^{2} |x-1| dx? —
1
22.
Evaluate \int_{0}^{\pi/2} \frac{\sin(x)}{\sin(x)+\cos(x)} dx. —
\pi/4
23.
Evaluate \int_{-1}^{1} x^3 e^{x^2} dx. —
0
24.
Using the property \int_{0}^{2a} f(x) dx = 0 if f(2a-x) = -f(x), evaluate \int_{0}^{2\pi} \cos(x) dx. —
0
25.
Evaluate \int_{0}^{\pi} \sin(2x) dx. —
0
26.
If f(x) is an odd function, what is the value of \int_{-a}^{a} f(x) dx? —
0
27.
Evaluate \int_{0}^{\pi/2} \frac{\cos(x)}{\sin(x)+\cos(x)} dx. —
\pi/4
28.
What is the value of \int_{0}^{\pi/2} \ln(\tan(x)) dx? —
0
29.
If f(x) is an odd function, then \int_{-a}^{a} f(x) dx is equal to: —
0
30.
What is the value of \int_{1}^{e} \frac{1}{x} dx? —
1
31.
Evaluate \int_{0}^{1} (2x+1) dx. —
3
32.
What is the value of \int_{0}^{1} (3x^2 - 2x + 1) dx? —
1
33.
Evaluate \int_{0}^{1} \frac{dx}{e^x+1}. —
1 - \frac{1}{2} \ln(2)
34.
Evaluate \int_{0}^{1} x^n dx. —
1/(n+1)
35.
What is the value of \int_{0}^{1} x(1-x)^n dx? —
1/(n+1)(n+2)
36.
What is the value of \int_{0}^{1} x \sin(\pi x) dx? —
1/\pi
37.
Evaluate \int_{0}^{1} (1-x)^{10} dx. —
1/11
38.
What is the value of \int_{0}^{\pi/2} \frac{\sin x \cos x}{\sin^4 x + \cos^4 x} dx? —
1/2 \arctan(1)
39.
What is the value of the definite integral \int_{0}^{1} x^2 dx? —
1/3
40.
What is the value of \int_{0}^{1} x \sqrt{1-x^2} dx? —
1/3
41.
Evaluate \int_{0}^{1} x \sqrt{1-x^4} dx. —
1/4
42.
Evaluate \int_{0}^{1} x^3 dx. —
1/4
43.
Evaluate \int_{0}^{\pi/2} \sin(2x) \cos(x) dx. —
2/3
44.
Which property is used to show that \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx? —
Property of Limits
45.
Which property is most useful for evaluating \int_{0}^{a} f(x) dx when f(x) exhibits symmetry about x=a/2? —
Property \int_{0}^{2a} f(x) dx = 2 \int_{0}^{a} f(x) dx if f(2a-x) = f(x)
46.
Evaluate \int_{0}^{1} (e^x + x) dx. —
e+1/2
47.
Evaluate \int_{0}^{1} x^2 e^x dx. —
e-2
48.
If \int_{0}^{a} f(x) dx = 0, what can be said about f(x)? —
f(x) is an odd function
49.
If \int_{0}^{a} f(x) dx = \int_{0}^{a} f(a-x) dx, what does this imply about the function f(x) in relation to the interval [0, a]? —
f(x) is symmetric about x=a/2
50.
If \int_{a}^{b} f(x) dx = \int_{a}^{b} f(a+b-x) dx, which property is being applied? —
Symmetry Property of Definite Integrals