إجابات كتاب التمارين

إجابات كتاب التمارين

التكامل غير المحدود

أجد كلاً من التكاملات الآتية:

$\int (4 x + 2) d x$ (1)

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

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

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

$\int (6 x^{2} - 4 x) d x$ (3)

$\int (6 x^{2} - 4 x) d x = 2 x^{3} - 2 x^{2} + C$

$\int (3 - x - 2 x^{5}) d x$ (4)

$\int (3 - x - 2 x^{5}) d x = 3 x - \frac{1}{2} x^{2} - \frac{1}{3} x^{6} + C$

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

$\int (x^{- 2} + x^{\frac{5}{2}}) d x = - x^{- 1} + \frac{2}{7} x^{\frac{7}{2}} + C$

$\int (3 x^{2} - \frac{2}{x^{2}}) d x$ (6)

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

$\int (3 x^{- 2} + 6 x^{- 1 / 2} + x - 4) d x$ (7)

$\int (3 x^{- 2} + 6 x^{- \frac{1}{2}} + x - 4) d x = - 3 x^{- 1} + 12 x^{\frac{1}{2}} + \frac{1}{2} x^{2} - 4 x + C$

$\int (10 x^{4} + 8 x^{- 3}) d x$ (8)

$\int (10 x^{4} + 8 x^{- 3}) d x = 2 x^{5} - 4 x^{- 2} + C$

$\int (\frac{2}{x^{3}} - 3 \sqrt{x}) d x$ (9)

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

$\int (8 x^{3} + 6 x - \frac{4}{\sqrt{x}}) d x$ (10)

$\begin{aligned} \int (8 x^{3} + 6 x - \frac{4}{\sqrt{x}}) d x & = \int (8 x^{3} + 6 x - 4 x^{- \frac{1}{2}}) d x \\ = 2 x^{4} + 3 x^{2} - 8 x^{\frac{1}{2}} + C \\ = 2 x^{4} + 3 x^{2} - 8 \sqrt{x} + C \end{aligned}$

$\int (\frac{7}{x^{2}} + \sqrt[3]{x^{4}}) d x$ (11)

$\begin{aligned} \int (\frac{7}{x^{2}} + \sqrt[3]{x^{4}}) d x & = \int (7 x^{- 2} + x^{\frac{4}{3}}) d x \\ = - 7 x^{- 1} + \frac{3}{7} x^{\frac{7}{3}} + C \\ = - \frac{7}{x} + \frac{3}{7} \sqrt[3]{x^{7}} + C \end{aligned}$

$\int (\frac{x^{2}}{3} + \frac{3}{x^{2}}) d x$ (12)

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

أجد كلاً من التكاملات الآتية:

$\int \frac{4 + 2 \sqrt{x}}{x^{2}} d x$ (13)

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

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

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

$\int \frac{x^{2} - 1}{x^{2}} d x$ (15)

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

$\int x \sqrt{x} d x$ (16)

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

$\int \frac{x^{2} - 1}{x - 1} d x$ (17)

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

$\int x^{2} (1 - x^{3}) d x$ (18)

$\begin{aligned} \int x^{2} (1 - x^{3}) d x & = \int (x^{2} - x^{5}) d x \\ = \frac{1}{3} x^{3} - \frac{1}{6} x^{6} + C \end{aligned}$

$\int (x + 4)^{2} d x$ (19)

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

$\int \frac{5 - x}{x^{5}} d x$ (20)

$\begin{aligned} \int \frac{5 - x}{x^{5}} d x & = \int (\frac{5}{x^{5}} - \frac{x}{x^{5}}) d x \\ = \int (5 x^{- 5} - x^{- 4}) d x \\ = - \frac{5}{4} x^{- 4} + \frac{1}{3} x^{- 3} + C \\ = - \frac{5}{4 x^{4}} + \frac{1}{3 x^{3}} + C \end{aligned}$

$\int \frac{x^{2} + 2 x + 1}{x + 1} d x$ (21)

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

$\int x (x + 1)^{2} d x$ (22)

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

$\int \frac{(x + 3)^{2}}{\sqrt{x}} d x$ (23)

$\begin{aligned} \int \frac{(x + 3)^{2}}{\sqrt{x}} d x & = \int \frac{x^{2} + 6 x + 9}{\sqrt{x}} d x \\ = \int (\frac{x^{2}}{\sqrt{x}} + \frac{6 x}{\sqrt{x}} + \frac{9}{\sqrt{x}}) d x \\ = \int (x^{\frac{3}{2}} + 6 x^{\frac{1}{2}} + 9 x^{\frac{1}{2}}) d x \\ = \frac{2}{5} x^{\frac{5}{2}} + 4 x^{\frac{3}{2}} + 18 x^{\frac{1}{2}} + C \end{aligned}$

$\int (x - 5) (x + 5) d x$ (24)

$\begin{aligned} \int (x - 5) (x + 5) d x & = \int (x^{2} - 25) d x \\ = \frac{1}{3} x^{3} - 25 x + C \end{aligned}$

إعداد : شبكة منهاجي التعليمية

09 / 02 / 2023

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