أتدرب وأحل المسائل

أتدرب وأحل المسائل

التكامل غير المحدود

أجد اقتراناً أصلياً لكلّ من الاقترانات الآتية:

(1) f(x) = x⁷

$\begin{aligned} f (x) = x^{7} \\ G (x) = \frac{1}{8} x^{8} + C \end{aligned}$

(2) f(x) = -2x⁶

$\begin{aligned} f (x) = - 2 x^{6} \\ G (x) = - \frac{2}{7} x^{7} + C \end{aligned}$

(3) f(x) = -10

$\begin{aligned} f (x) = - 10 \\ G (x) = - 10 x + C \end{aligned}$

(4) f(x) = 8x

$\begin{aligned} f (x) = 8 x \\ G (x) = 4 x^{2} + C \end{aligned}$

أجد كلاً من التكاملات الآتية:

(5) $\int 6 x d x$

$\int 6 x d x = 3 x^{2} + C$

(6) $\int (7 x - 5) d x$

$\int (7 x - 5) d x = \frac{7}{2} x^{2} - 5 x + C$

(7) $\int (3 - 4 x) d x$

$\int (3 - 4 x) d x = 3 x - 2 x^{2} + C$

(8) $\int \frac{10}{\sqrt{x}} d x$

$\begin{aligned} \int \frac{10}{\sqrt{x}} d x & = \int 10 x^{- \frac{1}{2}} d x \\ = 20 x^{\frac{1}{2}} + C \\ = 20 \sqrt{x} + C \end{aligned}$

(9) $\int {2 x}^{3 / 2} d x$

$\int 2 x^{\frac{3}{2}} d x = \frac{4}{5} x^{\frac{5}{2}} + C$

(10) $\int (2 x^{4} - 5 x + 10) d x$

$\int (2 x^{4} - 5 x + 10) d x = \frac{2}{5} x^{5} - \frac{5}{2} x^{2} + 10 x + C$

(11) $\int (2 x^{3} - 2 x) d x$

$\int (2 x^{3} - 2 x) d x = \frac{1}{2} x^{4} - x^{2} + C$

(12) $\int (\frac{3}{\sqrt[3]{x}} - \sqrt{x^{3}}) d x$

$\begin{aligned} \int (\frac{3}{\sqrt[3]{x}} - \sqrt{x^{3}}) d x & = \int (3 x^{- \frac{1}{3}} - x^{\frac{3}{2}}) d x \\ = \frac{9}{2} x^{\frac{2}{3}} - \frac{2}{5} x^{\frac{5}{2}} + C = \frac{9}{2} \sqrt[3]{x^{2}} - \frac{2}{5} \sqrt{x^{5}} + C \end{aligned}$

(13) $\int (\frac{1}{x^{2}} - \frac{1}{x^{3}}) d x$

$\begin{aligned} \int (\frac{1}{x^{2}} - \frac{1}{x^{3}}) d x & = \int (x^{- 2} - x^{- 3}) d x \\ = - x^{- 1} + \frac{1}{2} x^{- 2} + C = - \frac{1}{x} + \frac{1}{2 x^{2}} + C \end{aligned}$

أجد كلاً من التكاملات الآتية:

(14) $\int \frac{4 x^{3} - 2}{x^{3}} d x$

$\begin{aligned} \int \frac{4 x^{3} - 2}{x^{3}} d x & = \int (\frac{4 x^{3}}{x^{3}} - \frac{2}{x^{3}}) d x \\ = \int (4 - 2 x^{- 3}) d x \\ = 4 x + x^{- 2} + C = 4 x + \frac{1}{x^{2}} + C \end{aligned}$

(15) $\int \frac{2 x + 8}{\sqrt{x}} d x$

$\begin{aligned} \int \frac{2 x + 8}{\sqrt{x}} d x = & \int (\frac{2 x}{\sqrt{x}} + \frac{8}{\sqrt{x}}) d x \\ = \int (2 x^{\frac{1}{2}} + 8 x^{- \frac{1}{2}}) d x \\ = \frac{4}{3} x^{\frac{3}{2}} + 16 x^{\frac{1}{2}} + C \\ = \frac{4}{3} \sqrt{x^{3}} + 16 \sqrt{x} + C \end{aligned}$

(16) $\int (x - 1)^{2} d x$

$\begin{aligned} \int (x - 1)^{2} d x & = \int (x^{2} - 2 x + 1) d x \\ = \frac{1}{3} x^{3} - x^{2} + x + C \end{aligned}$

(17) $\int \frac{x^{3} + 8}{x + 2} d x$

$\begin{aligned} \int \frac{x^{3} + 8}{x + 2} d x & = \int \frac{(x + 2) (x^{2} - 2 x + 4)}{x + 2} d x \\ = \int (x^{2} - 2 x + 4) d x \\ = \frac{1}{3} x^{3} - x^{2} + 4 x + C \end{aligned}$

(18) $\int \sqrt{x} (x - 1) d x$

$\begin{aligned} \int \sqrt{x} (x - 1) d x & = \int (x^{\frac{3}{2}} - x^{\frac{1}{2}}) d x \\ = \frac{2}{5} x^{\frac{5}{2}} - \frac{2}{3} x^{\frac{3}{2}} + C = \frac{2}{5} \sqrt{x^{5}} - \frac{2}{3} \sqrt{x^{3}} + C \end{aligned}$

(19) $\int (2 x - 3) (3 x - 1) d x$

$\begin{aligned} \int (2 x - 3) (3 x - 1) d x & = \int (6 x^{2} - 2 x - 9 x + 3) d x \\ = \int (6 x^{2} - 11 x + 3) d x \\ = 2 x^{3} - \frac{11}{2} x^{2} + 3 x + C \end{aligned}$