ĐGTD ĐH Bách khoa - Tư duy Toán học - Tích phân

$\underset{a}{\overset{c}{\int }}f\left(x\right)d\text{x}=\underset{a}{\overset{b}{\int }}f\left(x\right)d\text{x}+\underset{b}{\overset{c}{\int }}f\left(x\right)d\text{x}$

$\underset{a}{\overset{b}{\int }}f\left(x\right)d\text{x}=\underset{a}{\overset{c}{\int }}f\left(x\right)d\text{x}-\underset{b}{\overset{c}{\int }}f\left(x\right)d\text{x}$

$\underset{a}{\overset{b}{\int }}f\left(x\right)d\text{x}=\underset{b}{\overset{a}{\int }}f\left(x\right)d\text{x}+\underset{a}{\overset{c}{\int }}f\left(x\right)d\text{x}$

$\underset{a}{\overset{b}{\int }}cf\left(x\right)d\text{x}=-\text{c}\underset{b}{\overset{a}{\int }}f\left(x\right)d\text{x}$

1

2

3

0

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=-\underset{b}{\overset{a}{\int }}f\left(x\right)dx$

$\underset{a}{\overset{b}{\int }}kdx=k\left(b-a\right)$

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx+\underset{b}{\overset{c}{\int }}f\left(x\right)dx=\underset{a}{\overset{c}{\int }}f\left(x\right)dx$

$\underset{a}{\overset{b}{\int }}f\left(x\right)dx=\underset{b}{\overset{a}{\int }}f\left(-x\right)dx$

I = 12

I = 48

I = 8

I = 3

$\underset{0}{\overset{1}{\int }}f\left(x\right)\text{d}x=-1.$

$\underset{0}{\overset{1}{\int }}f\left(x\right)\text{d}x=1.$

$\underset{0}{\overset{1}{\int }}f\left(x\right)\text{d}x=2.$

$\underset{0}{\overset{1}{\int }}f\left(x\right)\text{d}x=-\text{\hspace{0.17em}2}.$

-1

2

$-\frac{55}{32}$

-2

$\underset{0}{\overset{1}{\int }}f\left(x\right)dx\ge 0$

$\underset{0}{\overset{1}{\int }}g\left(x\right)dx\mathit{\le }0$

$\underset{0}{\overset{1}{\int }}g\left(x\right)dx\ge \underset{0}{\overset{1}{\int }}f\left(x\right)dx$

$\underset{0}{\overset{1}{\int }}f\left(x\right)dx\le 0$

29

5

19

40

32

34

46

40

-15

7

15

-7

$\underset{1}{\overset{4}{\int }}\left[f\left(x\right)+g\left(x\right)\right]dx=10$

$\underset{3}{\overset{4}{\int }}f\left(x\right)dx=-5$

$\underset{3}{\overset{4}{\int }}f\left(x\right)dx=5$

$\underset{1}{\overset{4}{\int }}\left[4f\left(x\right)-2g\left(x\right)\right]dx=-2$

$\frac{\pi }{4}$

$\frac{\pi }{2}$

$\frac{3\pi }{2}$

$\frac{\pi }{3}$

b = 1 hoặc b = −1

b = 0 hoặc b = 1

b = 0 hoặc b = 5

b = 1 hoặc b = 5

$\frac{3}{2}$

$-\frac{3}{2}$

$\frac{5}{2}$

$-\frac{5}{2}$

3a-b<12

a+2b=13

a-b>2

$a^2+b^2=41$

30

31

5

10

$\frac{7}{12}$

$\frac{7}{4}$

$\frac{9}{4}$

$\frac{3}{4}$

30

18

20

36

$\frac{19}{3}$

$\frac{32}{3}$

$\frac{16}{3}$

$\frac{21}{2}$

1

-1

0

2

$f\left(x\right)=\cos 3x$

$f\left(x\right)=\sin 3x$

$f\left(x\right)=\cos \left(\frac{x}{4}+\frac{\pi }{2}\right)$

$f\left(x\right)=\sin \left(\frac{x}{4}+\frac{\pi }{2}\right)$

$\frac{1}{2}\ln \frac{1}{3}$

$2\ln 3$

$\frac{1}{2}\ln 3$

$2\ln \frac{1}{3}$

12,5

9

11

10

$\frac{2\ln 2}{3}$

$-\frac{2\ln 2}{3}$

-2ln2

2ln2

I.J = 8

$I.J=\frac{32}{5}$

$I-J=\frac{128}{7}$

$I+J=\frac{64}{9}$

$4\sqrt{2}$

$3\sqrt{2}$

$\sqrt{2}$

$-\sqrt{2}$

0

$\frac{64}{3}$

7

12,5

$\frac{1}{3}+6\ln \frac{4}{3}$

$\frac{1}{2}+6\ln \frac{4}{3}$

$\frac{1}{2}-\ln \frac{4}{3}$

$\frac{1}{2}+\ln \frac{4}{3}$

3

5

6

4

4

3

5

6

$\frac{{\pi }^2-6}{18}$

$\frac{{\pi }^2-3}{32}$

$\frac{3{\pi }^2-16}{64}$

$\frac{3{\pi }^2-6}{112}$

$\frac{73}{6}$

$\frac{133}{9}$

$\frac{182}{9}$

$\frac{91}{6}$

$a\in \left[-\frac{3}{2};-\frac{1}{2}\right]$

$a\in \left[0;1\right]$

$a\in \left(-\infty ;-\frac{3}{2}\right]\cup \left[-\frac{1}{2};+\infty \right)$

$a\le 0$

$\frac{1}{3}.$

$\frac{5}{6}.$

$\frac{1}{2}$

$\frac{3}{2}$

$k\pi (k\in Z)$

$\frac{\pi }{4}+k\pi (k\in Z)$

$\frac{\pi }{2}+k\pi (k\in Z)$

$2k\pi (k\in Z)$

0

$-4043\sqrt{2}$

$2\sqrt{2}$

$4034\sqrt{2}$

4

$\frac{5}{3}$

5

$\frac{4}{3}$