قيمة \( \int \sin(2x - \pi)dx \) هي:
A
\( 2\cos(2x - \pi) + c \)
B
\( \frac{1}{2}\cos(2x - \pi) + c \)
C
\( -2\cos(2x - \pi) + c \)
D
\( -\frac{1}{2}\cos(2x - \pi) + c \)
Explanation
The correct answer is \( -\frac{1}{2}\cos(2x - \pi) + c \).
قيمة \( \int_1^e (2x - \frac{1}{x})dx \) هي:
A
e2
B
e2 - 2
C
\( \frac{1}{2}e^2 - 1 \)
D
\( \frac{1}{2}e^2 - 2 \)
Explanation
The correct answer is e2 - 2 .
قيمة \( \int_{-1}^1 (2 - |x|)dx \) هي:
Explanation
The correct answer is 0.
قيمة \( \int_0^1 e^{-x}dx \) هي:
A
\( \frac{1}{e} - 1 \)
B
\( -\frac{1}{e} \)
C
\( \frac{1}{e} \)
D
\( 1 - \frac{1}{e} \)
Explanation
The correct answer is \( 1 - \frac{1}{e} \).
قيمة \( \int_0^{\pi/2} \cos^2 x dx \) هي:
A
\( \frac{\pi}{2} \)
B
\( -\frac{\pi}{2} \)
C
\( \frac{\pi}{4} \)
D
\( -\frac{\pi}{4} \)
Explanation
The correct answer is \( \frac{\pi}{4} \).
ناتج \( \int 6e^{3x-1}dx \) هو:
A
-3e3x-1 + c
B
3e3x-1 + c
C
2e3x-1 + c
D
-2e3x-1 + c
Explanation
The correct answer is 3e3x-1 + c .
ناتج \( \int \cos(7 - 5x)dx \) هو:
A
\( -\frac{1}{5}\sin(7 - 5x) + c \)
B
\( -\frac{1}{7}\sin(7 - 5x) + c \)
C
\( \frac{1}{5}\sin(7 - 5x) + c \)
D
\( \frac{1}{7}\sin(7 - 5x) + c \)
Explanation
The correct answer is \( -\frac{1}{5}\sin(7 - 5x) + c \).
قيمة \( \int_0^{\pi/4} \tan^2 x dx \) هي:
A
\( -1 - \frac{\pi}{4} \)
B
\( -1 + \frac{\pi}{4} \)
C
\( 1 + \frac{\pi}{4} \)
D
\( 1 - \frac{\pi}{4} \)
Explanation
The correct answer is \( 1 - \frac{\pi}{4} \).
قيمة \( \int_0^6 \frac{6x}{x^2 + 1} dx \) هي:
A
6 ln(e2 + 1)
B
-6 ln(e2 + 1)
C
-3 ln(e2 + 1)
D
3 ln(e2 + 1)
Explanation
The correct answer is 6 ln(e2 + 1) .
قيمة \( \int_4^6 (5 + |3 - x|)dx \) هي:
Explanation
The correct answer is 6.
ناتج \( \int 2e^{1-2x}dx \) هو:
A
4e1-2x + c
B
-4e1-2x + c
C
e1-2x + c
D
-e1-2x + c
Explanation
The correct answer is 4e1-2x + c .
ناتج \( \int \cot^2 x dx \) هو:
A
\( \cot x - x + c \)
B
tan x - x + c
C
\( -\cot x - x + c \)
D
-tan x - x + c
Explanation
The correct answer is \( \cot x - x + c \).
إذا كان \( \int_3^k (2x - 3)dx = 4 \) فإن قيمة k هي:
Explanation
The correct answer is 4.
قيمة \( \int_1^2 \frac{x^2 - 6}{2x} dx \) هي:
A
\( \frac{3}{4} + \ln 8 \)
B
1 + ln 8
C
\( \frac{3}{4} - \ln 8 \)
D
1 - ln 8
Explanation
The correct answer is \( \frac{3}{4} + \ln 8 \).
قيمة \( \int_0^2 e^{2x} dx \) هي:
A
e4 - 1
B
e4 - 2
C
2e4 - 2
D
\( \frac{1}{2}e^4 - \frac{1}{2} \)
Explanation
The correct answer is e4 - 1 .
قيمة \( \int_{-4}^4 (4 - |x|)dx \) هي:
Explanation
The correct answer is 16.
إذا كان \( \int_a^{2a} (2 + 2x)dx = 1, a > 0 \) فجد قيمة الثابت a :
A
a = 0.25
B
a = 0.5
C
a = 1
D
a = 2
Explanation
The correct answer is a = 0.25.
إذا كان \( f'(x) = \cos^2 x \) يمثل ميل المماس لمنحنى الاقتران f فجد قاعدة الاقتران f الذي يمر منحناه بنقطة الأصل:
A
\( f(x) = \frac{x}{2} + \frac{\sin 2x}{4} \)
B
f(x) = sin2 x
C
f(x) = cos x + x
D
f(x) = tan x
Explanation
The correct answer is \( f(x) = \frac{x}{2} + \frac{\sin 2x}{4} \).
إذا كان \( \int_a^{2a} \frac{1 + 4x}{x} dx = \ln 32, a > 0 \) فجد قيمة الثابت a :
A
a = 2
B
a = 4
C
a = 1
D
a = 8
Explanation
The correct answer is a = 2.
إذا كان \( \int_1^e \frac{2x^2 - k}{x} dx = e^2 - 5 \) فجد قيمة الثابت k :
A
k = 2
B
k = 5
C
k = 1
D
k = 0
Explanation
The correct answer is k = 2.
إذا كان \( f'(x) = \sin 2x \) يمثل ميل المماس لمنحنى الاقتران f فجد قاعدة الاقتران f الذي يمر منحناه بالنقطة (0, 2) :
A
\( f(x) = -\frac{1}{2}\cos 2x + 2.5 \)
B
\( f(x) = \frac{1}{2}\sin 2x + 2 \)
C
\( f(x) = \frac{1}{2}\cos 2x + 2 \)
D
\( f(x) = -\frac{1}{2}\sin 2x + 2 \)
Explanation
The correct answer is \( f(x) = -\frac{1}{2}\cos 2x + 2.5 \).
أوجد ناتج التكامل الآتي: \( \int \frac{1}{\sqrt{e^x}} dx \)
A
\( 2\sqrt{e^x} + C \)
B
\( \sqrt{e^x} + C \)
C
\( \frac{1}{2}\sqrt{e^x} + C \)
D
e^x + C
Explanation
The correct answer is \( 2\sqrt{e^x} + C \).
أوجد ناتج التكامل الآتي: \( \int \sec^2(2x - 1)dx \)
A
\( \frac{1}{2}\tan(2x - 1) + C \)
B
tan(2x - 1) + C
C
2tan(2x - 1) + C
D
\( \frac{1}{2}\sec(2x - 1) + C \)
Explanation
The correct answer is \( \frac{1}{2}\tan(2x - 1) + C \).
أوجد ناتج التكامل الآتي: \( \int_0^2 |x^3 - 1|dx \)
A
تعتمد الإجابة على تقسيم التكامل عند x = 1 : \( \int_0^1 (1 - x^3)dx + \int_1^2 (x^3 - 1)dx = \frac{15}{4} \)
B
\( \int_0^2 (x^3 - 1)dx = \frac{7}{2} \)
C
\( \int_0^2 |x^3 - 1|dx = 2 \)
D
\( \int_0^2 (1 - x^3)dx = -6 \)
Explanation
The correct answer is تعتمد الإجابة على تقسيم التكامل عند x = 1 : \( \int_0^1 (1 - x^3)dx + \int_1^2 (x^3 - 1)dx = \frac{15}{4} \).
أوجد ناتج التكامل الآتي: \( \int_0^{\pi/4} (\sec^2 x + \cos 4x)dx \)
Explanation
The correct answer is 1.
أوجد ناتج التكامل الآتي: \( \int_0^{\pi/3} (\sin(2x + \frac{\pi}{3}) - 1 + \cos 2x)dx \)
A
\( [-\frac{1}{2}\cos(2x + \frac{\pi}{3}) - x + \frac{1}{2}\sin 2x]_0^{\pi/3} \)
B
\( \frac{1}{2}\sin(2x + \frac{\pi}{3}) + x - \frac{1}{2}\cos 2x \)
C
\( [-\frac{1}{2}\sin(2x + \frac{\pi}{3}) + x + \frac{1}{2}\cos 2x]_0^{\pi/3} \)
D
\( \frac{1}{2}\cos(2x + \frac{\pi}{3}) + x - \frac{1}{2}\sin 2x \)
Explanation
The correct answer is \( [-\frac{1}{2}\cos(2x + \frac{\pi}{3}) - x + \frac{1}{2}\sin 2x]_0^{\pi/3} \).
أوجد ناتج التكامل الآتي: \( \int_0^{\pi/8} \sin 2x \cos 2x dx \)
A
\( \int_0^{\pi/8} \frac{1}{2}\sin 4x dx = [-\frac{1}{8}\cos 4x]_0^{\pi/8} = \frac{1}{8} \)
B
\( \int_0^{\pi/8} \sin 2x dx = [-\frac{1}{2}\cos 2x]_0^{\pi/8} \)
C
\( \int_0^{\pi/8} \cos 2x dx = [\frac{1}{2}\sin 2x]_0^{\pi/8} \)
D
\( \int_0^{\pi/8} \sin x \cos x dx = [-\frac{1}{2}\cos 2x]_0^{\pi/8} \)
Explanation
The correct answer is \( \int_0^{\pi/8} \frac{1}{2}\sin 4x dx = [-\frac{1}{8}\cos 4x]_0^{\pi/8} = \frac{1}{8} \).
أوجد ناتج التكامل الآتي: \( \int_0^3 (x - 5^x)dx \)
A
\( [\frac{1}{2}x^2 - \frac{5^x}{\ln 5}]_0^3 = \frac{9}{2} - \frac{125}{\ln 5} + \frac{1}{\ln 5} \)
B
\( \frac{1}{3}x^3 - \frac{5^x}{5} = 9 - 25 + 1 \)
C
[x2 - 5^x]0 3 = 6 - 120
D
\( [\frac{1}{2}x^2 + \frac{5^x}{\ln 5}]_0^3 = \frac{9}{2} + \frac{125}{\ln 5} - \frac{1}{\ln 5} \)
Explanation
The correct answer is \( [\frac{1}{2}x^2 - \frac{5^x}{\ln 5}]_0^3 = \frac{9}{2} - \frac{125}{\ln 5} + \frac{1}{\ln 5} \).
