∫[0 to π] sin(x) dx = ?
A
2
B
0
C
1
D
π
Analysis & Theory
∫ sin(x) dx = -cos(x). Evaluating from 0 to π: [-cos(x)]₀^π = (-(-1)) - (-1) = 2.
∫[0 to π] cos(x) dx = ?
A
0
B
2
C
π
D
-2
Analysis & Theory
∫ cos(x) dx = sin(x). Evaluating from 0 to π: sin(π) - sin(0) = 0 - 0 = 0.
∫[0 to 1] x^2 dx = ?
A
1/3
B
1/2
C
1/4
D
2/3
Analysis & Theory
∫ x^2 dx = x^3/3. Evaluating: (1^3/3 - 0) = 1/3.
If f(x) is even, then ∫[-a to a] f(x) dx = ?
A
2∫[0 to a] f(x) dx
B
0
C
∫[0 to a] f(x) dx
D
-∫[0 to a] f(x) dx
Analysis & Theory
By property of even functions, ∫[-a to a] f(x) dx = 2∫[0 to a] f(x) dx.
If f(x) is odd, then ∫[-a to a] f(x) dx = ?
A
0
B
2∫[0 to a] f(x) dx
C
∫[0 to a] f(x) dx
D
-2∫[0 to a] f(x) dx
Analysis & Theory
By property of odd functions, ∫[-a to a] f(x) dx = 0.
∫[0 to π/2] sin(x) dx = ?
A
1
B
0
C
2
D
π/2
Analysis & Theory
∫ sin(x) dx = -cos(x). Evaluating: (-cos(π/2)) - (-cos(0)) = 0 - (-1) = 1.
∫[0 to π/2] cos(x) dx = ?
A
1
B
0
C
π/2
D
2
Analysis & Theory
∫ cos(x) dx = sin(x). Evaluating: sin(π/2) - sin(0) = 1 - 0 = 1.
∫[0 to a] f(x) dx + ∫[0 to a] f(a-x) dx = ?
A
a f(a/2)
B
2∫[0 to a] f(x) dx
C
∫[0 to a] f(x) dx
D
af(x)
Analysis & Theory
Property: ∫[0 to a] f(x) dx = ∫[0 to a] f(a-x) dx. So sum = 2∫[0 to a] f(x) dx.
∫[0 to π/2] ln(sin(x)) dx = ?
A
-(π/2) ln(2)
B
0
C
π/2
D
-1
Analysis & Theory
Property: ∫[0 to π/2] ln(sin(x)) dx = ∫[0 to π/2] ln(cos(x)) dx. Result = -(π/2) ln(2).
∫[0 to 2a] f(x) dx, where f(x) is periodic with period 2a = ?
A
2∫[0 to a] f(x) dx
B
0
C
a f(a)
D
∫[0 to a] f(x) dx
Analysis & Theory
If f(x) has period 2a, then ∫[0 to 2a] f(x) dx = 2∫[0 to a] f(x) dx.