إجابات كتاب التمارين

تكامل اقترانات خاصة

أجد كلاً من التكاملات الآتية:

(1) $\int 4e^{-5x} dx$

$\int 4e^{-5x }dx=-\frac{4}{5}{e}^{-5x}+C$

(2) $\int (\sin 2x-\cos 2x) dx$

$\int (\sin 2x-\cos 2x) dx=-\frac{1}{2}\cos 2x-\frac{1}{2}\sin 2x+C$

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

$\int {\cos }^2 2x dx=\frac{1}{2}\int (1+\cos 4x) dx=\frac{1}{2}x+\frac{1}{8}\sin 4x+C$

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

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

(5) $\int (\frac{\cos x}{{\sin }^2 x}-2e^x) dx$

$\int (\cot x\csc x-2e^x) dx=-\csc x-2e^x+C$

(6) $\int (3\cos 3x-{\tan }^2 x) dx$

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

(7) $\int \cos x(1+{\csc }^2 x) dx$

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

(8) $\int \frac{x^2+x-4}{x+2} dx$

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

(9) $\int \frac{1}{\sqrt{e^x}} dx$

$\int \frac{1}{\sqrt{e^x}} dx=\int e^{-\frac{1}{2}x} dx=-2e^{-\frac{1}{2}x}+C$

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

$\int (\frac{1}{{\cos }^2 x}+\frac{1}{x^2}) dx=\int ({\sec }^2 x+x^{-2}) dx=\tan x-\frac{1}{x}+C$

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

$\int \frac{x^2-2x}{x^3-3x^2} dx=\frac{1}{3}\int \frac{3x^2-6x}{x^3-3x^2} dx=\frac{1}{3}\ln |x^3-3x^2|+C$

(12) $\int \ln e^{\cos x} dx$

$\int \ln e^{\cos x} dx=\int \cos x dx=\sin x+C$

(13) $\int {\sin }^2 \frac{x}{2} dx$

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

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

$\int \frac{3}{2x-1} dx=\frac{3}{2}\int \frac{2}{2x-1} dx=\frac{3}{2}\ln |2x-1|+C$

(15) $\int \frac{3-2\cos \frac{1}{2}x}{{\sin }^2 \frac{1}{2}x} dx$

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