كويز تفاعلي: اختبر نفسك
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يتناول هذا الاختبار موضوع التكامل بالأجزاء، وهو أحد التقنيات الأساسية في حساب التفاضل والتكامل للمرحلة الثانوية المتقدمة. يركز الاختبار على تطبيق القاعدة \(\int u dv = uv - \int v du\) لإيجاد تكاملات الدوال اللوغاريتمية، والأسية، والمثلثية، والدوال العكسية، بالإضافة إلى التكاملات المحدودة واستخدام الجداول والقوانين المختصرة.
يرجى الانتباه إلى أن المعلم قام بإعداد الأسئلة فقط، ولم يقم بإعداد الإجابات أو الشروحات المرفقة. وقد تم توليد الإجابات باستخدام تقنيات الذكاء الاصطناعي، لذلك قد تتضمن بعض الأخطاء أو عدم الدقة.
Evaluate \(\int x e^{2x} dx\)
A
\(\frac{1}{2} x e^{2x} - \frac{1}{2} e^{2x} + c\)
B
\(\frac{1}{2} x e^{2x} - \frac{1}{2} e^{2x} + c\)
C
\(\frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} + c\)
D
\(\frac{1}{2} x^2 e^x - \frac{1}{4} e^x + c\)
Explanation
The correct answer is \(\frac{1}{2} x e^{2x} - \frac{1}{4} e^{2x} + c\).
Evaluate \(\int_{1}^{2} x e^{x} dx\)
A
1
B
e2 + 1
C
e2 - 1
D
e2
Explanation
The correct answer is e2 .
Evaluate \(\int \ln(2x) dx\)
A
\(x \ln 2x - \int dx + c\)
B
\(\frac{\ln 2x}{x} - \int dx + c\)
C
\(2x \ln x - \int dx + c\)
D
x ln 2x - x2 + c
Explanation
The correct answer is \(x \ln 2x - \int dx + c\).
Evaluate \(\int_{1}^{10} \ln 2x dx\)
A
10 ln 20 - ln 2 - 9
B
10 ln 10 - 9
C
10(ln 20 - ln 2) - 9
D
10 ln 20 - ln 2 + 9
Explanation
The correct answer is 10 ln 20 - ln 2 - 9.
Evaluate \(\int x^2 \ln x dx\)
A
\(\frac{x^3}{3} \ln x - \frac{1}{9} x^3 + c\)
B
\(\frac{x^3}{3} \ln x - \frac{1}{3} x^3 + c\)
C
\(\frac{x^3}{3} \ln x + \frac{1}{9} x^3 + c\)
D
\(\frac{x^3}{3} \ln x + \frac{1}{3} x^3 + c\)
Explanation
The correct answer is \(\frac{x^3}{3} \ln x - \frac{1}{9} x^3 + c\).
Evaluate \(\int_{0}^{\pi} 2x \cos x dx\)
A
\((2x \sin x - 2 \cos x) \big|_0^\pi\)
B
\(2x \sin x - \int_{0}^{\pi} 2 \sin x dx\)
C
\((2x \sin x - 2x \cos x) \big|_0^\pi\)
D
\(2x \sin x \big|_0^\pi - \int_{0}^{\pi} 2 \sin x dx\)
Explanation
The correct answer is \(2x \sin x \big|_0^\pi - \int_{0}^{\pi} 2 \sin x dx\).
Evaluate \(\int_{0}^{\pi} 2x \sin x dx\)
Explanation
The correct answer is \(2\pi\).
Evaluate \(\int \tan^{-1} x dx\)
A
\(x \sin^{-1} x + \sqrt{1 - x^2} + c\)
B
\(x \tan^{-1} x - \frac{1}{2} \ln(1 + x^2) + c\)
C
\(\frac{x^2 \tan^{-1} x - x + \tan^{-1} x}{2} + c\)
D
\(x \cos^{-1} x - \sqrt{1 - x^2} + c\)
Explanation
The correct answer is \(x \tan^{-1} x - \frac{1}{2} \ln(1 + x^2) + c\).
Evaluate \(\int x \sec^2 x dx\)
A
\(x \cos^{-1} x - \sqrt{1 - x^2} + c\)
B
\(-x \cot x + \ln(\sin x) + c\)
C
-x cos x + sin x + c
D
x tan x + ln(cos x) + c
Explanation
The correct answer is x tan x + ln(cos x) + c .
Evaluate \(\int (x + 7) e^{2x} dx\)
A
\(\frac{1}{2} (x + 7) e^{2x} + \frac{1}{4} e^{2x} + c\)
B
\(-\frac{1}{2} (x + 7) e^{2x} - \frac{1}{4} e^{2x} + c\)
C
\(\frac{1}{2} (x + 7) e^{2x} - \frac{1}{4} e^{2x} + c\)
D
\(-\frac{1}{2} (x + 7) e^{2x} + \frac{1}{4} e^{2x} + c\)
Explanation
The correct answer is \(\frac{1}{2} (x + 7) e^{2x} - \frac{1}{4} e^{2x} + c\).
Evaluate \(\int x \sin 2x dx\)
A
\(-x \cos 2x + \frac{1}{2} \sin 2x + c\)
B
\(\frac{x}{2} \cos 2x - \frac{1}{4} \sin 2x + c\)
C
\(-\frac{x}{2} \cos 2x + \frac{1}{4} \sin 2x + c\)
D
\(\frac{x}{2} \cos 2x + \frac{1}{4} \sin 2x + c\)
Explanation
The correct answer is \(-\frac{x}{2} \cos 2x + \frac{1}{4} \sin 2x + c\).
Evaluate \(\int x \sin x \cos x dx\)
A
\(-\frac{x \cos(2x)}{4} + \frac{\sin(2x)}{8} + c\)
B
\(-\frac{x \cos x}{2} + \frac{\sin x}{4} + c\)
C
\(\frac{x \sin(2x)}{4} + \frac{\cos(2x)}{8} + c\)
D
\(-\frac{x \cos(2x)}{2} + \frac{\sin(2x)}{4} + c\)
Explanation
The correct answer is \(-\frac{x \cos(2x)}{4} + \frac{\sin(2x)}{8} + c\).
Evaluate \(\int 3x e^{-x} dx\)
A
-3e-x (x - 1) + c
B
3e-x (x + 1) + c
C
-3e-x (x + 1) + c
D
3e^x(x - 1) + c
Explanation
The correct answer is -3e-x (x + 1) + c .
Evaluate \(\int x^2 e^x dx\)
A
x2 e^x - x e^x + e^x + c
B
x2 e^x - 2x e^x + 2e^x + c
C
x2 e^x + 2x e^x + 2e^x + c
D
x2 e^x - x e^x + c
Explanation
The correct answer is x2 e^x - 2x e^x + 2e^x + c .
Evaluate \(\int e^{\sqrt{x}} dx\)
A
\(2\sqrt{x} e^{\sqrt{x}} - 2 e^{\sqrt{x}} + c\)
B
\(2\sqrt{x} e^{\sqrt{x}} + 2 e^{\sqrt{x}} + c\)
C
\(2\sqrt{x} e^{\sqrt{x}} - 2 e^{x} + c\)
D
\(\frac{2}{\sqrt{x}} e^{\sqrt{x}} + c\)
Explanation
The correct answer is \(2\sqrt{x} e^{\sqrt{x}} - 2 e^{\sqrt{x}} + c\).
Evaluate \(\int \cos \sqrt{x} dx\)
A
\(2\sqrt{x} \sin \sqrt{x} - 2 \cos \sqrt{x} + c\)
B
\(2\sqrt{x} \sin \sqrt{x} + 2 \cos \sqrt{x} + c\)
C
\(2\sqrt{x} \cos \sqrt{x} - 2 \sin \sqrt{x} + c\)
D
\(-2\sqrt{x} \sin \sqrt{x} + \cos \sqrt{x} + c\)
Explanation
The correct answer is \(2\sqrt{x} \sin \sqrt{x} + 2 \cos \sqrt{x} + c\).
أي من التكاملات التالية يمكن إيجاده بالتكامل بالأجزاء؟ Which of the following integral can be found using integration by parts?
A
\(\int x \sin x^2 dx\)
B
\(\int \sqrt{x} dx\)
C
\(\int (x^2 - 5)^2 dx\)
D
\(\int x^2 \sin x dx\)
Explanation
The correct answer is \(\int x^2 \sin x dx\).
اعتماداً على المعلومات التالية: \(f(0) = 1, f'(0) = 5, f(2) = 3, f'(2) = 7\)، أوجد قيمة التكامل \(\int_{0}^{2} 3x f''(x) dx\).
Explanation
The correct answer is 36.
إذا كان \(\int x^2 \cos 3x dx = f(x) - \frac{2}{3} \int x \sin 3x dx\)، فإن f(x) =
A
\(\frac{2}{3} x \sin(3x)\)
B
\(\frac{1}{3} x^2 \sin(3x)\)
C
\(\frac{2}{3} x \cos(3x)\)
D
\(\frac{1}{3} x \sin(3x) - \frac{2}{3} \cos 3x\)
Explanation
The correct answer is \(\frac{1}{3} x^2 \sin(3x)\).
إذا كان \(\int \sin^{-1} x dx = x \sin^{-1} x + \int f(x) dx\)، فإن f(x) =
A
\(-x \sqrt{1 - x^2}\)
B
\(x \sqrt{1 - x^2}\)
C
\(\frac{-x}{\sqrt{1 - x^2}}\)
D
\(\frac{x}{\sqrt{1 - x^2}}\)
Explanation
The correct answer is \(\frac{-x}{\sqrt{1 - x^2}}\).
إذا كان \(\int \cos^{-1} x dx = x \cos^{-1} x + \int f(x) dx\)، فإن f(x) =
A
\(-x \sqrt{1 - x^2}\)
B
\(x \sqrt{1 - x^2}\)
C
\(-\frac{1}{\sqrt{1 - x^2}}\)
D
\(\frac{x}{\sqrt{1 - x^2}}\)
Explanation
The correct answer is \(\frac{x}{\sqrt{1 - x^2}}\).
أوجد \(\int f(x) \cos(nx) dx =\)
A
\(\frac{1}{n} f(x) \sin(nx) - \frac{1}{n} \int f'(x) \sin(nx) dx\)
B
\(\frac{1}{n} f(x) \cos(nx) - \frac{1}{n} \int f'(x) \cos(nx) dx\)
C
\(n f(x) \cos(nx) + \frac{1}{n} \int f'(x) \sin(nx) dx\)
D
\(n f(x) \sin(nx) - \frac{1}{n} \int f'(x) \cos(nx) dx\)
Explanation
The correct answer is \(\frac{1}{n} f(x) \sin(nx) - \frac{1}{n} \int f'(x) \sin(nx) dx\).
أوجد قيمة a, b, c في المعادلة: \(\int \cos^5 x dx = \frac{1}{a} \cos^b x \sin x + \frac{b}{a} \int \cos^c x dx\)
A
a = 5, b = 4, c = 3
B
a = 4, b = 3, c = 2
C
a = 3, b = 4, c = 5
D
a = 5, b = 4, c = 5
Explanation
The correct answer is a = 5, b = 4, c = 3.
اعتمد على الجدول التالي، إذا كان \(\int_{1}^{3} f(x) g'(x) dx = 8\)، فإن \(\int_{1}^{3} f'(x) g(x) dx =\)
Explanation
The correct answer is -4.
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