رياضيات فصل ثاني

التوجيهي علمي

الشرح الملخص أوراق العمل حل اسئلة الدرس النتاجات

الدرس الأول : تكاملات اقترانات خاصة

صيغ تكاملات اقترانات أسية :

أتحقق من فهمي صفحة 10 :

أجد كلاّ من التكاملات الآتية :

$a \int (5 x^{2} - 3 e^{7 x}) d x S o l u t i o n : \int (5 x^{2} - 3 e^{7 x}) d x = \int 5 x^{2} d x - \int 3 e^{7 x} d x = \frac{5 x^{3}}{3} - \frac{3 e^{7 x}}{7} + c$

$b \int_{0}^{l n 3} (8 e^{x}) d x S o l u t i o n : \int_{0}^{l n 3} (8 e^{x}) d x = 8 e^{x} \ln 3 0 = 8 e^{l n 3} - 8 e^{0} = 8 \times 3 - 8 \times 1 = 16$

$c \int \sqrt{e^{1 - x}} d x S o l u t i o n : \int \sqrt{e^{1 - x}} d x = \int e^{\frac{1}{2} (1 - x)} d x = \frac{e^{\frac{1}{2} (1 - x)}}{- \frac{1}{2}} + c = - 2 e^{\frac{1}{2} (1 - x)} + c$

$d \int (3^{x} + 2 \sqrt{x}) d x S o l u t i o n : \int (3^{x} + 2 \sqrt{x}) d x = \int (3^{x} + 2 x^{\frac{1}{2}}) d x = \frac{3^{x}}{l n 3} + \frac{4 x^{\frac{3}{2}}}{3} + c$

صيغ تكاملات اقترانات مثلثية:

أتحقق من فهمي صفحة 12 :

أجد كلاّ من التكاملات الآتية :

$a \int c o s (3 x - π) d x S o l u t i o n : \int c o s (3 x - π) d x = \frac{1}{3} s i n (3 x - π) + c$

$b \int (c s c^{2} (5 x) + e^{2 x}) d x S o l u t i o n : \int ({c s c}^{2} (5 x) + e^{2 x}) d x = \int {c s c}^{2} (5 x) d x + \int e^{2 x} d x = - \frac{1}{5} c o t (5 x) + \frac{1}{2} e^{2 x} + c$

$c \int_{0}^{\frac{π}{3}} (s i n 2 x - c o s 4 x) d x s o l u t i o n : \int_{0}^{\frac{π}{3}} (s i n 2 x - c o s 4 x) d x = \int_{0}^{\frac{π}{3}} s i n 2 x d x - \int_{0}^{\frac{π}{3}} c o s 4 x d x = - \frac{1}{2} c o s 2 x - \frac{1}{4} s i n 4 x \frac{π}{3} 0 = - \frac{1}{2} (c o s \frac{2 π}{3} - c o s 0) - \frac{1}{4} (s i n \frac{4 π}{3} - s i n 0) = - \frac{1}{2} (- \frac{1}{2} - 1) - \frac{1}{4} (- \frac{\sqrt{3}}{2} - 0) = \frac{3}{4} + \frac{\sqrt{3}}{8} = \frac{6 + \sqrt{3}}{8}$

المتطابقات المثلثية والتكامل :

أتحقق من فهمي صفحة 14:

أجد كلاّ من التكاملات الآتية :

$a \int c o s^{4} x d x S o l u t i o n : c o s^{2} x = \frac{1 + c o s 2 x}{2} {(c o s^{2} x)}^{2} = {(\frac{1 + c o s 2 x}{2})}^{2} c o s^{4} x = \frac{1}{4} (1 + 2 c o s 2 x + c o s^{2} 2 x) = \frac{1}{4} (1 + 2 c o s 2 x + \frac{1 + c o s 4 x}{2}) c o s^{4} x = \frac{3}{8} + \frac{1}{2} c o s 2 x + \frac{1}{8} (c o s 4 x) \int c o s^{4} x d x = \int (\frac{3}{8} + \frac{1}{2} c o s 2 x + \frac{1}{8} (c o s 4 x)) d x = \frac{3}{8} x + \frac{1}{4} s i n 2 x + \frac{1}{32} s i n 4 x + c$

$b \int_{0}^{\frac{π}{6}} s i n 3 x s i n x d x S o l u t i o n : s i n 3 x s i n x = \frac{1}{2} (c o s (3 x - x) - c o s (3 x + x)) = \frac{1}{2} (c o s (2 x) - c o s (4 x)) \int_{0}^{\frac{π}{6}} s i n 3 x s i n x d x = \frac{1}{2} \int_{0}^{\frac{π}{6}} (c o s 2 x - c o s 4 x) d x = \frac{1}{2} (\frac{1}{2} s i n 2 x - \frac{1}{4} s i n 4 x) \frac{π}{6} 0 = \frac{1}{4} (s i n \frac{π}{3} - s i n 0) - \frac{1}{8} (s i n \frac{2 π}{3} - s i n 0) = \frac{1}{4} (\frac{\sqrt{3}}{2}) - \frac{1}{8} (\frac{\sqrt{3}}{2}) = \frac{\sqrt{3}}{16}$

$c \int \frac{1}{1 + c o s x} d x S o l u t i o n : \int \frac{1}{1 + c o s x} d x \frac{1}{1 + c o s x} = \frac{1}{1 + c o s x} \times \frac{1 - c o s x}{1 - c o s x} = \frac{1 - c o s x}{1 - c o s^{2} x} = \frac{1 - c o s x}{s i n^{2} x} = \frac{1}{s i n^{2} x} - \frac{c o s x}{s i n^{2} x} = c s c^{2} x - c s c x c o t x \int \frac{1}{1 + c o s x} d x = \int (c s c^{2} x - c s c x c o t x) d x = - c o t x + c s c x + c$

تكاملات ينتج منها اقتران لوغرتمي :

أتحقق من فهمي صفحة 16:

أجد كلاّ من التكاملات الآتية :

$a \int (s i n x - \frac{5}{x}) d x S o l u t i o n : \int (s i n x - \frac{5}{x}) d x = \int \sin x d x - 5 \int \frac{1}{x} d x = - c o s x - 5 l n x + c$

$b \int \frac{5}{3 x + 2} d x S o l u t i o n : \int \frac{5}{3 x + 2} d x = \frac{5}{3} \int \frac{3}{3 x + 2} d x = \frac{5}{3} l n (3 x + 2) + c$

$c \int \frac{x^{2} - 7 x + 2}{x^{2}} d x S o l u t i o n : \int \frac{x^{2} - 7 x + 2}{x^{2}} d x = \int (\frac{x^{2}}{x^{2}}) d x - \int (\frac{7 x}{x^{2}}) d x + \int (\frac{2}{x^{2}}) d x = \int d x - 7 \int \frac{1}{x} d x + 2 \int x^{- 2} d x = x - 7 l n x + 2 \frac{x^{- 1}}{- 1} + c = x - 7 l n x - \frac{2}{x} + c$

$d \int \frac{2 x + 3}{x^{2} + 3 x} d x S o l u t i o n : \int \frac{2 x + 3}{x^{2} + 3 x} d x = l n (x^{2} + 3 x) + c$

$e \int \frac{s i n 2 x}{1 + c o s 2 x} d x S o l u t i o n : \int \frac{s i n 2 x}{1 + c o s 2 x} d x = \int \frac{s i n 2 x}{1 + c o s 2 x} d x = - \frac{1}{2} l n (1 + c o s 2 x) + c$

$f \int c o t x d x S o l u t i o n : \int c o t x d x = \int \frac{c o s x}{s i n x} d x = l n (s i n x) + c$

$g \int \frac{e^{x}}{e^{x} + 7} d x S o l u t i o n : \int \frac{e^{x}}{e^{x} + 7} d x = l n (e^{x} + 7) + c$

$h \int c s c x d x S o l u t i o n : c s c x = c s c x \times \frac{c s c x - c o t x}{c s c x - c o t x} = \frac{c s c^{2} x - c s c x c o t x}{c s c x - c o t x} \int c s c x d x = \int \frac{c s c^{2} x - c s c x c o t x}{c s c x - c o t x} d x = - l n (c s c x - c o t x) + c$

أتحقق من فهمي صفحة 17:

أجد $\int \frac{x^{2} + x + 1}{x + 1} d x$ :

$S o l u t i o n : \int \frac{x^{2} + x + 1}{x + 1} d x = \int (x + \frac{1}{x + 1}) d x = \frac{1}{2} x^{2} + l n (x + 1) + c$

تكامل الاقترانات المتشعبة :

أتحقق من فهمي صفحة 19:

إذا كان $f (x) = {\begin{cases} x + 1 & , x < 1 \\ 2 x & , x \geq 1 \end{cases}$ ، فأجد قيمة $\int_{- 1}^{3} f (x) d x$ :

$S o l u t i o n : \int_{- 1}^{3} f (x) d x = \int_{- 1}^{1} (x + 1) d x + \int_{1}^{3} 2 x d x = \frac{1}{2} x^{2} + x 1 - 1 + x^{2} 3 1 = \frac{1}{2} ({(1)}^{2} - {(- 1)}^{2}) + (1 - - 1) + ({(3)}^{2} - {(1)}^{2}) = 0 + 2 + 8 = 10$

إذا كان $f (x) = | 1 - x |$ ، فأجد قيمة $\int_{- 2}^{2} f (x) d x$ :

$S o l u t i o n : \int_{- 2}^{2} | 1 - x | d x = \int_{- 2}^{1} (1 - x) d x + \int_{1}^{2} (x - 1) d x = x - \frac{1}{2} x^{2} 1 - 2 + \frac{1}{2} x^{2} - x 2 1 = (1 - - 2) - \frac{1}{2} (1 - 4) + \frac{1}{2} (4 - 1) - (2 - 1) = 3 + \frac{3}{2} + \frac{3}{2} - 1 = 5$

إذا كان $f (x) = | x^{2} - 1 |$ ، فأجد قيمة $\int_{- 4}^{0} f (x) d x$ :

$S o l u t i o n : \int_{- 4}^{0} | x^{2} - 1 | d x = \int_{- 4}^{- 1} (x^{2} - 1) d x + \int_{- 1}^{0} (1 - x^{2}) d x = \frac{1}{3} x^{3} - x - 1 - 4 + x - \frac{1}{3} x^{3} 0 - 1 = \frac{1}{3} (- 1 - - 64) - (- 1 - - 4) + (0 - - 1) - \frac{1}{3} (0 - - 1) = 21 - 3 + 1 - \frac{1}{3} = 19 - \frac{1}{3} = \frac{56}{3}$

تطبيقات التكامل الشرط الأوَّلي :

أتحقق من فهمي صفحة 20:

تلوّث: تسرَّب نفط من ناقلة بحرية مكوناً بقعة دائرية الشكل على سطح الماء .نصف قطرها $R (t)$ قدماً

بعد $(t)$ ثانية من بدء التسرُّب . إذا كان نصف قطر الدائرة يزداد بمعدَّل $R' (t) = \frac{21}{0.07 t + 5}$ .

فأجد $R (t)$ علماً $R (0) = 0$ .

$S o l u t i o n : R (t) = \int \frac{21}{0.07 t + 5} d t = \frac{21}{0.07} \int \frac{0.07}{0.07 t + 5} d t R (t) = \frac{21}{0.07} l n (0.07 t + 5) + c R (0) = \frac{21}{0.07} l n (0.07 (0) + 5) + c = 0 c = - \frac{21}{0.07} l n 5 R (t) = \frac{21}{0.07} l n (0.07 t + 5) - \frac{21}{0.07} l n 5 = \frac{21}{0.07} l n (\frac{0.07 t}{5} + 1)$

تطبيقات التكامل الحركة في مسارٍ مستقيم:

أتحقق من فهمي صفحة 23:

يتحرك جسيم في مسارٍ مستقيم وتعطى سرعته المتجهة بالاقتران $v (t) = 3 c o s t$ .

حيث $(t)$ الزمن بالثواني ، $v (t)$ السرعة المتجهة بالمتر لكل ثانية .

$a$ إذا بدء الجسيم حركته من نقطة الأصل ، فأجد موقع الجسم بعد $\frac{π}{6}$ ثانية من بدء الحركة .

$S o l u t i o n : s (t) = \int v (t) d t = \int 3 c o s t d t = 3 s i n t + c s (0) = 3 s i n 0 + c = 0 \Rightarrow c = 0 s (t) = 3 s i n t T h e n s (\frac{π}{6}) = 3 s i n (\frac{π}{6}) = \frac{3}{2} m$

$b$ أجد إزاحة الجسيم في الفترة $[0, 2 π]$ .

$S o l u t i o n : Δ s = s (t_{2}) - s (t_{1}) = \int_{t_{1}}^{t_{2}} v (t) d t = \int_{0}^{2 π} 3 c o s t d t = 3 s i n t 2 π 0 = 3 s i n 2 π - 3 s i n 0 = 0 \Rightarrow Δ s = 0$

$c$ أجد المسافة الكلية التي قطعها الجسيم في الفترة $[0, 2 π]$ .

$S o l u t i o n : s (t) = \int_{t 1}^{t_{2}} | v (t) | d t = \int_{0}^{2 π} | 3 c o s t | d t = 4 \int_{0}^{\frac{π}{2}} 3 c o s t d t = 4 s i n t \frac{π}{2} 0 = 4 s i n \frac{π}{2} - 4 s i n 0 = 4 m$

تمارين ومسائل صفحة 26 :

أجد كلاّ من التكاملات الآتية :

$1 \int (e^{2 x - 3} + \sqrt{x}) d x S o l u t i o n : \int (e^{2 x - 3} + \sqrt{x}) d x = \int (e^{2 x - 3} + x^{\frac{1}{2}}) d x = \frac{1}{2} e^{2 x - 3} + \frac{2}{3} x^{\frac{3}{2}} + c$

$2 \int (e^{0.5} - \frac{3}{e^{0.5}}) d x S o l u t i o n : \int (e^{0.5} - \frac{3}{e^{0.5}}) d x = (e^{0.5} - \frac{3}{e^{0.5}}) x + c$

$3 \int (4 s i n 5 x - 5 c o s 4 x) d x S o l u t i o n : \int (4 s i n 5 x - 5 c o s 4 x) d x = - \frac{4}{5} c o s 5 x - \frac{5}{4} s i n 4 x + c$

$4 \int (3 s e c x t a n x - \frac{2}{5 x}) d x S o l u t i o n : \int (3 s e c x t a n x - \frac{2}{5 x}) d x = 3 s e c x - \frac{2}{5} l n x + c$

$5 \int {(\sqrt{e^{x}} - \frac{1}{\sqrt{e^{x}}})}^{2} d x S o l u t i o n : \int {(\sqrt{e^{x}} - \frac{1}{\sqrt{e^{x}}})}^{2} d x = {(\sqrt{e^{x}} - \frac{1}{\sqrt{e^{x}}})}^{2} = {(\sqrt{e^{x}})}^{2} - 2 \sqrt{e^{x}} \times \frac{1}{\sqrt{e^{x}}} + {(\frac{1}{\sqrt{e^{x}}})}^{2} = e^{x} - 2 + e^{- x} \int {(\sqrt{e^{x}} - \frac{1}{\sqrt{e^{x}}})}^{2} d x = \int (e^{x} - 2 + e^{- x}) d x = e^{x} - 2 x + \frac{e^{- x}}{- 1} + c = e^{x} - 2 x - \frac{1}{e^{x}} + c$

$6 \int (s i n (5 - 3 x) + 2 + 4 x^{2}) d x S o l u t i o n : \int (s i n (5 - 3 x) + 2 + 4 x^{2}) d x = \frac{1}{3} c o s (5 - 3 x) + 2 x + \frac{4}{3} x^{3} + c$

$7 \int {(e^{x} + 1)}^{2} d x S o l u t i o n : {(e^{x} + 1)}^{2} = e^{2 x} + 2 e^{x} + 1 \int {(e^{x} + 1)}^{2} d x = \int (e^{2 x} + 2 e^{x} + 1) d x = \frac{1}{2} e^{2 x} + 2 e^{x} + x + c$

$8 \int (e^{4 - x} + s i n (4 - x) + c o s (4 - x)) d x S o l u t i o n : \int (e^{4 - x} + s i n (4 - x) + c o s (4 - x)) d x = - e^{4 - x} + c o s (4 - x) - s i n (4 - x) + c$

$9 \int \frac{x^{4} - 6}{2 x} d x S o l u t i o n : \int \frac{x^{4} - 6}{2 x} d x = \int (\frac{1}{2} x^{3} + \frac{3}{x}) d x = \frac{1}{8} x^{4} + 3 l n x + c$

$10 \int (3 s e c^{2} (3 x + 2) + \frac{5}{x}) d x S o l u t i o n : \int (3 s e c^{2} (3 x + 2) + \frac{5}{x}) d x = t a n (3 x + 2) + 5 l n x + c$

$11 \int \frac{e^{x} + 1}{e^{x}} d x S o l u t i o n : \int \frac{e^{x} + 1}{e^{x}} d x = \int \frac{e^{x}}{e^{x}} d x + \int \frac{1}{e^{x}} d x = x + \int e^{- x} d x = x - e^{- x} + c$

$12 \int \frac{e^{x}}{e^{x} + 4} d x S o l u t i o n : \int \frac{e^{x}}{e^{x} + 4} d x = l n (e^{x} + 4) + c$

$13 \int \frac{c o s 2 x}{s i n x c o s x + 4} d x S o l u t i o n : \int \frac{c o s 2 x}{s i n x c o s x + 4} d x = \int \frac{2 c o s 2 x}{2 s i n x c o s x + 8} d x = \int \frac{2 c o s 2 x}{s i n 2 x + 8} d x = l n (s i n 2 x + 8) + c$

$14 \int \frac{1}{5 - \frac{x}{3}} d x S o l u t i o n : \int \frac{1}{5 - \frac{x}{3}} d x = - 3 \int \frac{- 1}{15 - x} d x = - 3 l n | 15 - x | + c .$

$15 \int \frac{1}{1 - s i n x} d x S o l u t i o n : \frac{1}{1 - s i n x} \times \frac{1 + s i n x}{1 + s i n x} = \frac{1 + s i n x}{1 - s i n^{2} x} = \frac{1 + s i n x}{c o s^{2} x} = s e c^{2} x + \frac{s i n x}{c o s x} \times \frac{1}{c o s x} = s e c^{2} x + t a n x s e c x \int \frac{1}{1 - s i n x} d x = \int s e c^{2} x d x + \int t a n x s e c x d x = t a n x + s e c x + c$

$16 \int s e c^{2} x (1 + e^{x} c o s^{2} x) d x S o l u t i o n : s e c^{2} x (1 + e^{x} c o s^{2} x) = s e c^{2} x + e^{x} \frac{c o s^{2} x}{c o s^{2} x} \int s e c^{2} x (1 + e^{x} c o s^{2} x) d x = \int (s e c^{2} x + e^{x}) d x = t a n x + e^{x} + c$

$17 \int (\frac{2}{x} - 2^{x}) d x S o l u t i o n : \int (\frac{2}{x} - 2^{x}) d x = 2 l n x - \frac{2^{x}}{l n 2} + c$

$18 \int s i n 3 x c o s 2 x d x s o l u t i o n : s i n 3 x c o s 2 x = \frac{1}{2} (s i n (3 x + 2 x) + s i n (3 x - 2 x)) = \frac{1}{2} (s i n (5 x) + s i n (x)) \int s i n 3 x c o s 2 x d x = \frac{1}{2} \int (s i n (5 x) + s i n (x)) d x = \frac{1}{2} (- \frac{1}{5} \cos 5 x - \cos x) + c$

$19 \int \frac{2 x + 3}{3 x^{2} + 9 x - 1} d x S o l u t i o n : \int \frac{2 x + 3}{3 x^{2} + 9 x - 1} d x = \frac{1}{3} \int \frac{6 x + 9}{3 x^{2} + 9 x - 1} d x = \frac{1}{3} l n (3 x^{2} + 9 x - 1) + c$

$20 \int \frac{x^{2} + x + 1}{x^{2} + 1} d x S o l u t i o n : \int \frac{x^{2} + x + 1}{x^{2} + 1} d x = \int \frac{x^{2} + 1}{x^{2} + 1} d x + \frac{1}{2} \int \frac{2 x}{x^{2} + 1} d x = \int d x + \frac{1}{2} \int \frac{2 x}{x^{2} + 1} d x = x + \frac{1}{2} l n (x^{2} + 1) + c$

$21 \int (\frac{1 + c o s x}{s i n^{2} x} + s i n^{2} x c s c x) d x S o l u t i o n : \frac{1 + c o s x}{s i n^{2} x} + s i n^{2} x c s c x = \frac{1}{s i n^{2} x} + \frac{1}{s i n x} \times \frac{c o s x}{s i n x} + s i n^{2} x \times \frac{1}{s i n x} = c s c^{2} x + c s c x c o t x + s i n x \int (\frac{1 + c o s x}{s i n^{2} x} + s i n^{2} x c s c x) d x = \int c s c^{2} x d x + \int c s c x c o t x d x + \int s i n x d x = - c o t x - c s c x - c o s x + c$

$22 \int {(s e c x + t a n x)}^{2} d x S o l u t i o n : {(s e c x + t a n x)}^{2} = s e c^{2} x + 2 s e c x t a n x + t a n^{2} x = s e c^{2} x + 2 s e c x t a n x + s e c^{2} x - 1 = 2 s e c^{2} x + 2 s e c x t a n x - 1 \int {(s e c x + t a n x)}^{2} d x = 2 t a n x + 2 s e c x - x + c$

$23 \int \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} d x S o l u t i o n : \int \frac{e^{x} - e^{- x}}{e^{x} + e^{- x}} d x = l n (e^{x} + e^{- x}) + c$

$24 \int \frac{x^{2}}{x^{3} - 3} d x S o l u t i o n : \int \frac{x^{2}}{x^{3} - 3} d x = \frac{1}{3} \int \frac{3 x^{2}}{x^{3} - 3} d x = \frac{1}{3} l n | x^{3} - 3 | + c$

$25 \int (9 c o s^{2} x - s i n^{2} x - 6 s i n x c o s x) d x S o l u t i o n : 6 s i n x c o s x = \frac{6}{2} (s i n (x + x) + s i n (x - x)) = 3 (s i n (2 x) + s i n (0)) = 3 \sin 2 x 9 c o s^{2} x - s i n^{2} x = 9 c o s^{2} x + c o s^{2} x - 1 9 c o s^{2} x - s i n^{2} x = 10 c o s^{2} x - 1 \int (9 c o s^{2} x - s i n^{2} x - 6 s i n x c o s x) d x = = \int (10 c o s^{2} x - 1 - 3 s i n 2 x) d x = \int (10 (\frac{1}{2} (1 + \cos 2 x)) - 1 - 3 s i n 2 x) d x = \int (5 + 5 \cos 2 x - 1 - 3 s i n 2 x) d x = \int (4 + 5 c o s 2 x - 3 s i n 2 x) d x = 4 x + \frac{5}{2} \sin 2 x + \frac{3}{2} \cos 2 x + c$

$26 \int (c o s^{4} x - s i n^{4} x) d x S o l u t i o n : \int (c o s^{4} x - s i n^{4} x) d x (c o s^{4} x - s i n^{4} x) = (c o s^{2} x - s i n^{2} x) (c o s^{2} x + s i n^{2} x) = (c o s 2 x) (1) \int (c o s^{4} x - s i n^{4} x) d x = \int (c o s 2 x) d x = \frac{1}{2} s i n 2 x + c$

$27 \int_{0}^{π} 2 c o s (\frac{1}{2} x) d x S o l u t i o n : \int_{0}^{π} 2 c o s (\frac{1}{2} x) d x = 4 s i n (\frac{1}{2} x) π 0 = 4 s i n (\frac{π}{2}) - 4 s i n (\frac{0}{2}) = 4 - 0 = 4$

$28 \int_{0}^{2 π} | s i n x | d x S o l u t i o n : \int_{0}^{2 π} | s i n x | d x = 4 \int_{0}^{\frac{π}{2}} s i n x d x = - 4 c o s x \frac{π}{2} 0 = - 4 (c o s (\frac{π}{2}) - c o s (0)) = - 4 (0 - 1) = 4$

$29 \int_{\frac{π}{6}}^{\frac{π}{3}} 3 t a n^{2} x d x S o l u t i o n : \int_{\frac{π}{6}}^{\frac{π}{3}} 3 t a n^{2} x d x = 3 \int_{\frac{π}{6}}^{\frac{π}{3}} (s e c^{2} x - 1) d x = 3 (t a n (\frac{π}{3}) - t a n (\frac{π}{6})) - 3 (\frac{π}{3} - \frac{π}{6}) = 3 (\sqrt{3} - \frac{1}{\sqrt{3}}) - 3 (\frac{π}{6}) = \frac{6}{\sqrt{3}} - \frac{π}{2}$

$30 \int_{1}^{e} \frac{8 x}{x^{2} + 1} d x S o l u t i o n : \int_{1}^{e} \frac{8 x}{x^{2} + 1} d x = 4 \int_{1}^{e} \frac{2 x}{x^{2} + 1} d x = 4 l n (x^{2} + 1) e 1 = 4 l n (e^{2} + 1) - 4 l n (1^{2} + 1) = 4 l n (e^{2} + 1) - 4 l n (2) = l n {(\frac{e^{2} + 1}{2})}^{4}$

$31 \int_{0}^{\frac{π}{6}} s i n 3 x c o s x d x S o l u t i o n : s i n 3 x c o s x = \frac{1}{2} (s i n (3 x + x) + s i n (3 x - x)) = \frac{1}{2} (s i n (4 x) + s i n (2 x)) \int_{0}^{\frac{π}{6}} s i n 3 x c o s x d x = \frac{1}{2} \int_{0}^{\frac{π}{6}} (s i n (4 x) + s i n (2 x)) d x = - \frac{c o s 4 x}{8} - \frac{c o s 2 x}{4} \frac{π}{6} 0 = - \frac{c o s (\frac{2 π}{3}) - c o s 0}{8} - \frac{c o s (\frac{π}{3}) - c o s 0}{4} = - \frac{- \frac{1}{2} - 1}{8} - \frac{\frac{1}{2} - 1}{4} = \frac{3}{16} + \frac{1}{8} = \frac{5}{16}$

$32 \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{c o t^{2} x}{1 + c o t^{2} x} d x S o l u t i o n : \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{c o t^{2} x}{1 + c o t^{2} x} d x = \int_{\frac{π}{4}}^{\frac{π}{3}} \frac{c o t^{2} x}{c s c^{2} x} d x = \int_{\frac{π}{4}}^{\frac{π}{3}} c o s^{2} x d x = \frac{1}{2} \int_{\frac{π}{4}}^{\frac{π}{3}} (1 + c o s 2 x) d x = \frac{1}{4} s i n 2 x + \frac{x}{2} \frac{π}{3} \frac{π}{4} = \frac{1}{4} (s i n (\frac{2 π}{3}) - s i n (\frac{2 π}{4})$