Antiderivative ဟာ Grade 12 သင်ရိုးတွင် ပြဌာန်းလာမည့် သင်ခန်းစာတစ်ခုဖြစ်ပါတယ်။ သင်ခန်းစာရှင်းလင်းချက်များကို ဒီနေရာမှာ ရေခဲ့ဖူးပါတယ်။ အဆိုပါ post နှင့် ယှဉ်တွဲလေ့လာပြီး အောက်ပါ လေ့ကျင့်ခန်းများကို လေ့လာ လေ့ကျင့်ကြည့်နိုင်ပါတယ်။ စဉ်ဆက်မပြတ် လေ့လာသင်ယူနိုင်ကြပါစေ။
Rules of Integration
$\begin{array}{l} \text { 1. } \displaystyle\int k \ \mathrm{d}x=k x+c \\\\ \text { 2. } \displaystyle\int f^{\prime}(x) \ \mathrm{d}x=f(x)+c \\\\ \text { 3. } \displaystyle\int x^{n} \ \mathrm{d}x=\dfrac{x^{n+1}}{n+1}+c \\\\ \text { 4. } \displaystyle\int k f(x) \ \mathrm{d}x=k \displaystyle\int f(x) \ \mathrm{d}x\\\\ \text { 5. } \displaystyle\int\left[f(x) \pm g(x)\right] \ \mathrm{d}x = \displaystyle\int f(x) \ \mathrm{d}x \pm \displaystyle\int g(x) \ \mathrm{d}x \end{array}$ |
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- Integrate each of the following with respect to $x$.
(a) $x^3$
(b) $3\sqrt{x}$
(c) $\dfrac{2}{x^2}$
(d) $\dfrac{1}{2\sqrt{x}}$
(e) $(3x+5)\ \mathrm{d}x$ - Find each of the following indefinite integrals.
(a) $ \displaystyle\int (3x − 1)(x + 2)\ \mathrm{d}x$
(b) $ \displaystyle\int\left(3 x^{3}-4 \sqrt{x}+3\right) \ \mathrm{d}x$
(c) $ \displaystyle\int\left(6 x^{2}-\dfrac{4}{x^{2}}\right) \ \mathrm{d}x$
(d) $ \displaystyle\int\left(5-\dfrac{1}{\sqrt{x}}+\dfrac{1}{x^{3}}\right) \ \mathrm{d}x$
(e) $ \displaystyle\int \dfrac{x^{4}+5 x}{2 x^{3}} \ \mathrm{d}x$ - Find each of the following indefinite integrals.
(a) $\displaystyle\int \dfrac{3 x}{2 \sqrt[5]{x^{2}}} \mathrm{~d} x$
(b) $\displaystyle\int \dfrac{(3 x-1)^{2}}{5 x^{4}} \mathrm{~d} x$
(c) $\displaystyle\int \dfrac{3 x^{7}+x^{2}}{2 \sqrt[3]{x}} \mathrm{~d} x$
(d) $\displaystyle\int(x-3 \sqrt{x})^{2} \mathrm{~d} x$
(e) $\displaystyle\int(1+\sqrt[4]{x})(1-\sqrt[4]{x}) \mathrm{d} x$
(f) $\displaystyle\int\left(\sqrt[3]{x}+\dfrac{2}{\sqrt[3]{x}}\right)^{2} \mathrm{~d} x$ - The rate of change of A with respect to r is given by $\dfrac{dA}{dr}= 4r+7$. If $A = 12$ when $r = 1$,find $A$ in terms of $r$.
- Given that the gradient of a curve is $2x^2 + 7x$ and that the curve passes through the origin, determine the equation of the curve.
- A curve is such that $\dfrac{dy}{dx}=k\sqrt[3]{x}$ , where $k$ is a constant and that it passes through the points $(1, 4)$ and $(8, 16)$. Find the equation of the curve.
- The gradient of a curve at the point $(x, y)$ on the curve is given by $\dfrac{x^{2}-4}{x^{2}}$. Given that the curve passes through the point $(2,7)$, find the equation of the curve.
- A curve with $\dfrac{dy}{dx}=k x+3$, where $k$ is a constant, passes through the point $P(3,19)$. Given that the gradient of the normal to the curve at the point $P$ is $-\dfrac{1}{15}$, find
(i) the value of $k$,
(ii) the equation of the curve,
(iii) the coordinates of the turning point on the curve.
- The equation of a curve is such that $\dfrac{dy}{dx}=\dfrac{1}{(x-3)^{2}}+x .$ It is given that the curve passes through the point $(2,7)$. Find the equation of the curve.
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