إيجاد المشتقات العكسية والتكاملات غير المحددة: القواعد الأساسية والرموز
| English | العربية |
|---|---|
| indefinite integral/ɪnˈdefɪnət ˈɪntɪɡrəl/ | التكامل غير المحدد |
| constant of integration/ˈkɒnstənt ɒv ˌɪntɪˈɡreɪʃn/ | ثابت التكامل |
Running differentiation backward
- FTC Part 2 needs an antiderivative — so let's learn to find them.
- An antiderivative of $f$ is any function whose derivative is $f$; the whole family is the indefinite integral 不定积分.
- $\displaystyle\int f(x)\,dx$ (no limits) means "the general antiderivative of $f$."
- Every derivative rule, read backward, becomes an antiderivative rule.
The power rule for integration
- Reverse the Power Rule: add one to the exponent, then divide by the new exponent.
-
$$\int x^n\,dx = \frac{x^{n+1}}{n+1}+C\qquad(n\neq-1)$$
- Check: differentiating $\frac{x^{n+1}}{n+1}$ gives $x^n$. ✓
- (The case $n=-1$ is special: $\int x^{-1}\,dx=\ln|x|+C$.)
ما هو $\displaystyle\int x^3\,dx$؟
أضف واحداً للأس، واقسم عليه: $\tfrac{x^4}{4}+C$.
قاعدة القوة تفشل في $n=-1$. ما هي $\displaystyle\int x^{-1}\,dx$؟
$\int\tfrac1x\,dx=\ln|x|+C$.
Never forget $+C$
- Since the derivative of a constant is $0$, any constant can be added — so the antiderivative is a family.
- Always append the constant of integration 积分常数 $+C$ to an indefinite integral.
- Without limits there's no way to pin down which constant, so $+C$ stays.
- (In a definite integral the $+C$ cancels — that's the difference.)
عائلة مزاحة بـ +C
y = ax³ + d
جميع المشتقات العكسية لـ $f$ تختلف فقط بإزاحة عمودية $+C$ — نفس الميل في كل مكان.
يجب أن يتضمن التكامل غير المحدد ثابت التكامل $+C$.
المشتق العكسي هو عائلة؛ $+C$ جزء من الإجابة.
The standard antiderivatives
- Reverse the elementary derivatives you know:
- $\displaystyle\int\cos x\,dx=\sin x+C$; $\;\displaystyle\int\sin x\,dx=-\cos x+C$.
- $\displaystyle\int e^x\,dx=e^x+C$; $\;\displaystyle\int\frac1x\,dx=\ln|x|+C$.
- Combine with linearity to integrate any polynomial or sum term by term.
ما هو $\displaystyle\int \sin x\,dx$؟
$\int\sin x\,dx=-\cos x+C$ (انتبه للإشارة السالبة).
أوجد $\displaystyle\int (6x^2-4x+5)\,dx$.
قاعدة القوة العكسية حدًا بحد، زائد $C$.
حدد كل المشتقات العكسية الصحيحة.
$\int\cos x\,dx=\sin x+C$ (موجب)؛ الأخير خاطئ.
Two staples: always add $+C$ to an indefinite integral (it's part of the answer), and the power rule fails for $n=-1$ — $\int x^{-1}\,dx$ is $\ln|x|+C$, not $\frac{x^0}{0}$. Also mind the sign on $\int\sin x\,dx=-\cos x+C$ (a minus, mirroring $\cos$'s derivative).
Find $\displaystyle\int (6x^2 - 4x + 5)\,dx$.
- Term by term (reverse power rule): $6\cdot\dfrac{x^3}{3}=2x^3$; $\;-4\cdot\dfrac{x^2}{2}=-2x^2$; $\;5x$.
- $\displaystyle\int (6x^2-4x+5)\,dx = 2x^3-2x^2+5x+C$.
- Check by differentiating: $6x^2-4x+5$. ✓
An indefinite integral $\int f\,dx$ is the general antiderivative of $f$. The power rule for integration: $\int x^n\,dx=\frac{x^{n+1}}{n+1}+C$ for $n\neq-1$ (and $\int x^{-1}\,dx=\ln|x|+C$). Always add the constant of integration $+C$, and integrate sums term by term.