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Antiderivatives — Homework & Practice
Antiderivatives — Power Rule & Preparation
Homework and guided practice on antiderivatives, radicals, negative exponents, logarithms, and algebraic preparation before integration.
Practice the power rule, convert radicals to exponents, simplify fractions, expand brackets, and identify when the answer becomes a logarithm.
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مسائل الواجب المنزلي (Antiderivatives)
Homework, Examples, and Guided Practice — Antiderivatives
Question 1 / 15
Difficulty: 1/5
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⭐ القاعدة الذهبية (Golden Rule)
قبل التكامل جهّز الدالة أولًا:
- قاعدة القوة: زد الأس بمقدار 1 ثم اقسم على الأس الجديد.
- الجذور: حوّل الجذر إلى أس كسري قبل البدء.
- المقام: انقل الحدود من المقام إلى البسط بأس سالب متى أمكن.
- الضرب والقسمة: فك الأقواس أو وزع المقام على البسط أولًا إذا كان ذلك أسهل.
- اللوغاريتم: إذا ظهر الشكل \( \int \frac{1}{x}dx \) أو مشتقة المقام على المقام، فكر في \( \ln|x| \).
Core reminders
• Convert radicals to fractional exponents first.
• Move denominator terms to the top using negative powers when appropriate.
• Expand brackets before integrating if it makes the expression simpler.
• Divide the denominator into the numerator term-by-term when needed.
• Use \( \ln|x| \) for the special case \( \int \frac{1}{x}dx \).
• Move denominator terms to the top using negative powers when appropriate.
• Expand brackets before integrating if it makes the expression simpler.
• Divide the denominator into the numerator term-by-term when needed.
• Use \( \ln|x| \) for the special case \( \int \frac{1}{x}dx \).
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💡 Tips
⭐ Golden Rule
Prepare first, then integrate: convert radicals, move denominator powers up, simplify brackets, and apply the power rule carefully.
Quick formulas
\(\int x^n dx=\frac{x^{n+1}}{n+1}+C,\; n\neq -1\)
\(\int \frac{1}{x}dx=\ln|x|+C\)
\(\sqrt[n]{x^m}=x^{m/n}\)
\(x^a\cdot x^b=x^{a+b}\)
\(\frac{x^a}{x^b}=x^{a-b}\)
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