أتدرب وأحل المسائل

التكامل غير المحدود

أجد اقتراناً أصلياً لكلّ من الاقترانات الآتية:

(1) f(x) = x⁷

$\begin{array}{rl}& f\left(x\right)=x^7\\ & G\left(x\right)=\frac{1}{8}{x}^8+C\end{array}$

(2) f(x) = -2x⁶

$\begin{array}{rl}& f\left(x\right)=-2x^6\\ & G\left(x\right)=-\frac{2}{7}{x}^7+C\end{array}$

(3) f(x) = -10

$\begin{array}{rl}& f\left(x\right)=-10\\ & G\left(x\right)=-10x+C\end{array}$

(4) f(x) = 8x

$\begin{array}{rl}& f\left(x\right)=8x\\ & G\left(x\right)=4x^2+C\end{array}$

أجد كلاً من التكاملات الآتية:

(5) $\int 6x dx$

$\int 6x dx=3x^2+C$

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

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

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

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

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

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

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

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

(10) $\int (2x^4 - 5x + 10) dx$

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

(11) $\int (2x^3 - 2x) dx$

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

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

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

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

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

أجد كلاً من التكاملات الآتية:

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

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

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

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

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

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

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

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

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

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

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

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