إجابات كتاب التمارين

التكامل غير المحدود

أجد كلاً من التكاملات الآتية:

 (1)

 (2)

 (3)

 (4)

 (5)

 (6)

 (7)

 (8)

 (9)

 (10)

 (11)

 (12)

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)

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