11 integral
TRANSCRIPT
![Page 1: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/1.jpg)
Integral
Integral
![Page 2: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/2.jpg)
Integral adalah proses kebalikan dari diferensial.Apabila pada diferensial kita mulai dengan fungsi dan memproses menemukan koefisien diferensialnya, pada integral kita mulai dengan koefisien diferensial dan kemudian menentukan fungsi awalnya.
![Page 3: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/3.jpg)
Integral-integral Baku
1. xn dx = xn+1 + C {asalkan n≠ -1} n+1
2. dx = ln x + C
3. ex dx = ex + C
4. ekx dx = ekx + C k
5. ax dx = ax + C ln a
x1
![Page 4: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/4.jpg)
Integral-integral Baku
6. sin x dx = -cos x + C
7. cos x dx = sin x + C
8. sec2x dx = tan x + C
9. sinh x dx = cosh x + C
10. cosh x dx = sinh x + C
![Page 5: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/5.jpg)
Integral-integral Baku
11. √(1–x2) dx = sin-1x + C
12. √(1–x2) dx = cos-1x + C
13. 1+x2 dx = tan-1x + C
1 .
- 1 .
1 .
![Page 6: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/6.jpg)
Integral-integral Baku
14. √(x2+1) dx = sinh-1x + C
15. √(x2–1) dx = cosh-1x + C
16. (1–x2) dx = tanh-1x + C 1 .
1 .
1 .
![Page 7: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/7.jpg)
Latihan 1
1. e5x dx = ...
2. x7 dx = ...
3. √x dx = ...
4. sin x dx = ...
5. 2 sinh x dx = ...
6. dx = ...x 5 .
7. √(1–x2) dx = ...
8. 5x dx = ...
9. √(x2–1) dx = ...
10. 1+x2 dx = ...
1 .
1 .
1 .
![Page 8: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/8.jpg)
Fungsi dari suatu fungsi linear dalam x
Misal diberikan persamaan sebagai berikut:
Bagaimana kita dapat menyelesaikannya?
![Page 9: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/9.jpg)
x6 dx = x7 + C
(5x–4)6 dx = (5x–4)7 + C 35
cos x dx = sin x + C cos (2x+5) dx = sin (2x+5) + C
2 dx = ln x + C
2x+3 dx = ln (2x+3) + C 2
ex dx = ex + C e4x dx = e4x + C
4
1 .7
1 .x
1 .
Fungsi dari suatu fungsi linear dalam x
![Page 10: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/10.jpg)
Latihan 2
1. (2x–7)3 dx
2. cos (7x+2) dx
3. e5x+4 dx
4. sinh 7x dx
5. 4x+3 dx
6. 1+(2x)2 dx
7. sec2(3x+1) dx
8. sin (2x–5) dx
9. cosh (1+4x) dx
10. 35x dx 1 .
1 .
![Page 11: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/11.jpg)
Integrasi Parsial
Contoh
![Page 12: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/12.jpg)
Latihan 3
1
2
3
4
![Page 13: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/13.jpg)
Integral dalam bentuk f’(x) dan f’(x)∙f(x)
f(x)
Contoh: (2x+3) dx = ln (x2+3x–5) + C
(x2+3x–5)
3x2 dx = ln (x3–4) + C (x3–4)
f’(x) dx = ln {f(x)}
+C f(x)
![Page 14: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/14.jpg)
Integral dalam bentuk f’(x) dan f’(x)∙f(x)
f(x)
Contoh lain: cot x dx = cos x dx = ln sin x + C
sin x
tan x dx = sin x dx = (-sin x) cos x cos x
= – ln cos x + C
4x–8 dx = 2 2x–4 dx x2–4x+5 x2–4x+5
= 2 ln (x2–4x+5) + C
-
![Page 15: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/15.jpg)
Integral dalam bentuk f’(x) dan f’(x)∙f(x)
f(x)
Contoh: tan x ∙ sec2 x dx = tan x ∙ d(tan x)
= tan2 x + C2
ln x dx = ln x∙1dx= ln x ∙ d(ln x) x x
= (ln x)2 + C2
f’(x)∙f(x) dx = ½ {f(x)}2
+C
![Page 16: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/16.jpg)
Latihan 41. sec2x dx = ...
tan x
2. 2x+4 dx = ... x2+4x–1
3. sinh x dx = ... cosh x
4. 2x–6 dx = ... x2–6x+2
5. (x2+7x – 4)(2x+7) dx = ...
![Page 17: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/17.jpg)
uv = u dv dx + v du dx dx dx
Integrasi suatu perkalian
d (uv) = u dv + v du dx dx dx u dv dx = uv – v du dx dx dx
u dv = uv – v du
![Page 18: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/18.jpg)
Prioritas u:•ln x•xn
•ekx
![Page 19: 11 Integral](https://reader034.vdocuments.mx/reader034/viewer/2022042502/552f9c864a7959273b8b4613/html5/thumbnails/19.jpg)
Latihan 5
1. x ln x dx = ...
2. x3 e2x dx = ...