∫ ln(x) dx = ?
A
x ln(x) - x + C
B
ln(x^2)/2 + C
C
ln(x) + C
D
x/ln(x) + C
Analysis & Theory
Integration by parts: ∫ ln(x) dx = x ln(x) - x + C.
∫ 1/(x ln(x)) dx = ?
A
ln|ln(x)| + C
B
ln(x)/x + C
C
1/ln(x) + C
D
x ln(x) + C
Analysis & Theory
Let u = ln(x), then du = dx/x. So integral = ∫ 1/u du = ln|u| + C = ln|ln(x)| + C.
∫ (ln(x))^2 dx = ?
A
x((ln(x))^2 - 2 ln(x) + 2) + C
B
x (ln(x))^2 + C
C
ln(x)/x + C
D
ln(x^3)/3 + C
Analysis & Theory
Integration by parts gives ∫ (ln(x))^2 dx = x((ln(x))^2 - 2 ln(x) + 2) + C.
∫ (1 + ln(x)) dx = ?
A
x + x ln(x) + C
B
ln(x^2) + C
C
ln(x) + x + C
D
x ln(x) - x + C
Analysis & Theory
Split: ∫ 1 dx + ∫ ln(x) dx = x + (x ln(x) - x) = x ln(x) + C.
∫ (ln(x))/x dx = ?
A
(ln(x))^2/2 + C
B
ln(x^2) + C
C
1/x + C
D
ln|ln(x)| + C
Analysis & Theory
Let u = ln(x), then du = dx/x. So integral = ∫ u du = (u^2)/2 + C = (ln(x))^2/2 + C.
∫ sinh(x) dx = ?
A
cosh(x) + C
B
sinh(x) + C
C
tanh(x) + C
D
coth(x) + C
Analysis & Theory
Derivative of cosh(x) is sinh(x), hence ∫ sinh(x) dx = cosh(x) + C.
∫ cosh(x) dx = ?
A
cosh(x) + C
B
sinh(x) + C
C
tanh(x) + C
D
ln|cosh(x)| + C
Analysis & Theory
Derivative of sinh(x) is cosh(x), so ∫ cosh(x) dx = sinh(x) + C.
∫ sech^2(x) dx = ?
A
tanh(x) + C
B
coth(x) + C
C
cosh(x) + C
D
-tanh(x) + C
Analysis & Theory
Derivative of tanh(x) is sech^2(x), so ∫ sech^2(x) dx = tanh(x) + C.
∫ csch^2(x) dx = ?
A
-coth(x) + C
B
coth(x) + C
C
tanh(x) + C
D
-tanh(x) + C
Analysis & Theory
Derivative of coth(x) is -csch^2(x), hence ∫ csch^2(x) dx = -coth(x) + C.
∫ tanh(x) dx = ?
A
ln(cosh(x)) + C
B
ln(sinh(x)) + C
C
cosh(x) + C
D
sinh(x) + C
Analysis & Theory
Since d/dx[ln(cosh(x))] = tanh(x), we get ∫ tanh(x) dx = ln(cosh(x)) + C.