إجابات كتاب التمارين

التكامل غير المحدود

أجد كلاً من التكاملات الآتية:

$\int (4x+2)dx$ (1)

$\int (4x+2)dx=2x^2+2x+C$

$\int 2x^{-4}dx$ (2)

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

$\int (6x^2-4x)dx$ (3)

$\int (6x^2-4x)dx=2x^3-2x^2+C$

$\int (3-x-2x^5)dx$ (4)

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

$\int (x^{-2}+x^{5/2})dx$ (5)

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

$\int (3x^2-\frac{2}{x^2})dx$ (6)

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

$\int (3x^{-2}+6x^{-1/2}+x-4)dx$ (7)

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

$\int (10x^4+8x^{-3})dx$ (8)

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

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

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

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

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

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

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

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

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

أجد كلاً من التكاملات الآتية:

$\int \frac{4+2\sqrt{x}}{x^2}dx$ (13)

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

$\int \frac{4-x^2}{2+x}dx$ (14)

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

$\int \frac{x^2-1}{x^2}dx$ (15)

$\begin{array}{rl}\int \frac{x^2-1}{x^2}dx& =\int (\frac{x^2}{x^2}-\frac{1}{x^2})dx\\ & =\int (1-x^{-2})dx\\ & =x+x^{-1}+C\\ & =x+\frac{1}{x}+C\end{array}$

$\int x\sqrt{x}dx$ (16)

$\begin{array}{rl}\int x\sqrt{x}dx& =\int x^{\frac{3}{2}}dx\\ & =\frac{2}{5}{x}^{\frac{5}{2}}+C\\ & =\frac{2}{5}\sqrt{x^5}+C\end{array}$

$\int \frac{x^2-1}{x-1}dx$ (17)

$\begin{array}{rl}\int \frac{x^2-1}{x-1}dx& =\int \frac{(x-1)(x+1)}{x-1}dx\\ & =\int (x+1)dx\\ & =\frac{1}{2}{x}^2+x+C\end{array}$

$\int x^2(1-x^3)dx$ (18)

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

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

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

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

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

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

$\begin{array}{rl}\int \frac{x^2+2x+1}{x+1}dx& =\int \frac{(x+1)(x+1)}{x+1}dx\\ & =\int (x+1)dx\\ & =\frac{1}{2}{x}^2+x+C\end{array}$

$\int x(x+1{)}^{2}dx$ (22)

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

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

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

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

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