Integration using trigonometric identities - One Line Questions
1.
What is the integral of sin^3(x)cos^4(x) dx? —
-cos^5(x)/5 + cos^7(x)/7 + C
2.
What is the integral of sin(3x)cos(2x) dx? —
-cos(5x)/10 - cos(x)/2 + C
3.
What is the integral of cot(x)csc^2(x) dx? —
-cot^2(x)/2 + C
4.
What is the integral of cot^2(x) dx? —
-cot(x) - x + C
5.
What is the integral of csc(x)cot(x) dx? —
-csc(x) + C
6.
What is the integral of cos(x) / (1 - sin(x)) dx? —
-ln|1 - sin(x)| + C
7.
What is the integral of sin(x) / (1 + cos(x)) dx? —
-ln|1 + cos(x)| + C
8.
To integrate (sin(x) - cos(x))^2 dx, we first expand it to: —
1 - sin(2x)
9.
To integrate (sin(x) + cos(x))^2 dx, we first expand it to: —
1 + sin(2x)
10.
The integral of cos(x) / (1 + cos(x)) dx can be simplified by using the identity cos(x) = 2cos^2(x/2) - 1 and 1 + cos(x) = 2cos^2(x/2) to get the integral of: —
1/2 sec^2(x/2)
11.
The integral of cos(x)cos(3x) dx is found using the product-to-sum formula: —
2cos(A)cos(B) = cos(A-B) + cos(A+B)
12.
To integrate tan(x/2) dx, we use the substitution u = x/2, so dx = 2du, and the integral becomes: —
2ln|sec(x/2)| + C
13.
The integral of sin(6x)cos(4x) dx is found using the product-to-sum formula: —
2sin(A)cos(B) = sin(A+B) + sin(A-B)
14.
To integrate sin(x)cos(2x) dx, we use the identity: —
2sin(A)cos(B) = sin(A+B) + sin(A-B)
15.
Which trigonometric identity is used to integrate sin(ax)cos(bx)? —
2sin(A)cos(B) = sin(A+B) + sin(A-B)
16.
The integral of sin(x)sin(3x) dx is found using the product-to-sum formula: —
2sin(A)sin(B) = cos(A-B) - cos(A+B)
17.
The integral of sin(5x)sin(3x) dx uses the identity: —
2sin(A)sin(B) = cos(A-B) - cos(A+B)
18.
To integrate cos^4(x), we use the identity: —
cos^2(x) = (1 + cos(2x))/2
19.
Which trigonometric identity is most useful for integrating cos^2(x)? —
cos(2x) = 2cos^2(x) - 1
20.
The integral of cos^3(x)sin^4(x) dx can be solved by rewriting cos^3(x) as: —
cos(x)(1 - sin^2(x))
21.
The integral of (1 + cos(2x)) / (1 - cos(2x)) dx can be simplified using half-angle identities to the integral of: —
cot^2(x)
22.
To integrate cos(x) / (1 - cos(x)) dx, we can use the identity 1 - cos(x) = 2sin^2(x/2) and cos(x) = 1 - 2sin^2(x/2) to get the integral of: —
cot^2(x/2) - 1
23.
Which identity is used to integrate sec^2(x)? —
Integral of sec^2(x) is tan(x) + C
24.
To integrate sec(x)tan(x) dx, we use the identity: —
Integral of sec(x)tan(x) is sec(x) + C
25.
Which trigonometric identity is used to integrate sec^3(x)? —
Integration by parts using sec(x) * sec^2(x)
26.
What is the integral of sin(x) / (1 - cos(x)) dx? —
ln|1 - cos(x)| + C
27.
What is the integral of 1 / (sin(x)cos(x)) dx? —
ln|tan(x)| + C
28.
What is the integral of (1 + cos(x)) / sin(x) dx? —
ln|tan(x/2)| + C
29.
To integrate sec^4(x) dx, we can rewrite it as: —
sec^2(x) * sec^2(x) = (1 + tan^2(x))sec^2(x)
30.
The integral of sin^4(x) dx can be simplified by repeatedly applying the identity: —
sin^2(x) = (1 - cos(2x))/2
31.
What is the integral of cos(x)sin^3(x) dx? —
sin^4(x)/4 + C
32.
The integral of sin(x)cos(x) dx can be evaluated using which identity? —
sin(2x) = 2sin(x)cos(x)
33.
The integral of sin^2(x)cos^2(x) dx is found by using the identity: —
All of the above
34.
What is the integral of cos(5x)cos(3x) dx? —
sin(8x)/16 + sin(2x)/4 + C
35.
What is the integral of cos^3(x) dx? —
sin(x) - sin^3(x)/3 + C
36.
The integral of sin^3(x) dx can be simplified by rewriting sin^3(x) as: —
sin(x)(1 - cos^2(x))
37.
The integral of tan^2(x/2) dx can be evaluated by using the identity: —
tan^2(u) = sec^2(u) - 1
38.
To integrate tan^2(x), we typically use the identity: —
tan^2(x) = sec^2(x) - 1
39.
To integrate tan^3(x), we can rewrite it as tan(x) * tan^2(x) and use the identity: —
tan^2(x) = sec^2(x) - 1
40.
What is the integral of tan(x)sec^2(x) dx? —
tan^2(x)/2 + C
41.
What is the integral of tan^4(x) dx? —
tan^3(x)/3 - tan(x) + x + C
42.
What is the integral of (1 - cos(2x)) / (1 + cos(2x)) dx? —
tan(x) - x + C
43.
What is the integral of (1 - cos(x)) / (1 + cos(x)) dx? —
tan(x/2) - x + C
44.
What is the integral of sec(x) / (1 + sin(x)) dx? —
tan(x/2) + C
45.
To integrate cos(x) / (1 + sin(x)) dx, we can use the substitution: —
u = 1 + sin(x)
46.
The integral of sin(x)cos^3(x) dx can be solved by substituting: —
u = cos(x)
47.
What is the integral of (sin(2x) + cos(2x))^2 dx? —
x - cos(4x)/2 + C
48.
What is the integral of sin(x) / (1 + sin(x)) dx? —
x - 2tan(x/2) + C
49.
What is the integral of sin^2(x) with respect to x? —
x/2 - sin(2x)/4 + C
50.
What is the integral of cos^2(2x) dx? —
x/2 + sin(4x)/8 + C