The integral of x^n (n ≠ -1) with respect to x is?
A
x^(n+1)/(n+1) + C
B
n*x^(n-1) + C
C
ln(x) + C
D
1/(n+1) + C
Analysis & Theory
∫ x^n dx = (x^(n+1))/(n+1) + C, provided n ≠ -1.
∫ 1/x dx = ?
A
x + C
B
1/x + C
C
ln|x| + C
D
e^x + C
Analysis & Theory
The integral of 1/x is ln|x| + C.
∫ e^x dx = ?
A
x*e^x + C
B
e^x + C
C
ln(e^x) + C
D
1/e^x + C
Analysis & Theory
The derivative of e^x is itself, so ∫ e^x dx = e^x + C.
∫ cos(x) dx = ?
A
sin(x) + C
B
-sin(x) + C
C
cos(x) + C
D
-cos(x) + C
Analysis & Theory
Derivative of sin(x) is cos(x), hence ∫ cos(x) dx = sin(x) + C.
∫ sin(x) dx = ?
A
-cos(x) + C
B
cos(x) + C
C
sin(x) + C
D
-sin(x) + C
Analysis & Theory
Derivative of cos(x) is -sin(x), so ∫ sin(x) dx = -cos(x) + C.
∫ sec^2(x) dx = ?
A
sec(x) + C
B
tan(x) + C
C
cot(x) + C
D
cosec(x) + C
Analysis & Theory
Derivative of tan(x) is sec^2(x), hence ∫ sec^2(x) dx = tan(x) + C.
∫ 1/(1+x^2) dx = ?
A
tan^-1(x) + C
B
sin^-1(x) + C
C
cos^-1(x) + C
D
sec^-1(x) + C
Analysis & Theory
Derivative of tan^-1(x) is 1/(1+x^2), so ∫ 1/(1+x^2) dx = tan^-1(x) + C.
∫ 1/√(1-x^2) dx = ?
A
tan^-1(x) + C
B
sin^-1(x) + C
C
cos^-1(x) + C
D
ln|x| + C
Analysis & Theory
Derivative of sin^-1(x) is 1/√(1-x^2), hence ∫ 1/√(1-x^2) dx = sin^-1(x) + C.
∫ cos^2(x) dx can be simplified using?
A
tan^2(x) identity
B
cos(2x) identity
C
sin(2x) identity
D
No identity needed
Analysis & Theory
We use cos^2(x) = (1+cos(2x))/2 to integrate easily.
Which of the following is the most general representation of an indefinite integral?
A
It always includes + C
B
It never includes + C
C
It is unique
D
It is always finite
Analysis & Theory
Indefinite integrals always include + C (constant of integration) as they represent a family of functions.