integral e^x+2x^2 dx
1. integral e^x+2x^2 dx
[tex] \int\limits {e^x + 2x^2} \, dx \\ = e^x + \frac{2}{3} x^3 + c[/tex]Integral
[tex] \int\limits e^{x}+2x^{2} \, dx \\ =\boxed{\ e^{x} + \frac{2}{3}x^{3} + c} [/tex]
2. integral e^2x / e^2x + 3 =
Jika tidak terbaca lihat lampiran
[tex]\displaystyle misal:\\u=e^{2x}+3\\du=2e^{2x}dx\\\\~~~~~\int\frac{e^{2x}\,dx}{e^{2x}+3}\\\\=\frac{1}{2}\int\frac{2e^{2x}\,dx}{e^{2x}+3}\\\\=\frac{1}{2}\int\frac{du}{u}\\\\=\frac{1}{2}\ln(u)+C\\\\=\boxed{\frac{1}{2}\ln(e^{2x}+3)+C}[/tex]
3. Integral (e^(2x) + 1)^2 dx Trimakasih
semoga membantu .......
4. integral dari cosx e^2x dx
integral cosx e^2x dx
= sinx . 1/2 e^2x +c[tex]\int\limits^._. {cosx.e^{2x}} \, dx=[/tex]
form : [tex] \int\limits^._. {e^{bx} cos ax} \, dx= \frac{1}{a^{2}+b^{2}} e^{bx}(a. sinax+b.cosax)[/tex]
[tex]\int\limits^._. {e^{2x}.cosx} \, dx= e^{2x}\frac{1}{1^{2}+2^{2}}(sinx+2cosx) [/tex]
[tex]= e^{2x} \frac{1}{5}(sinx+2 cos x) [/tex]
[tex]= \frac{1}{5}e^{2x}sin x+ \frac{2}{5}e^{2x}cosx+c [/tex]
5. the result of integral (x-2)sin(2x-pi) dx is
#F
Parsial integral
∫ (x - 2) sin (2x - π) dx
= ∫ (x - 2) sin { - (π - 2x)} dx
= ∫ -(x- 2) sin (2x) dx
= ∫ - x sin 2x + 2 sin 2x dx
= -x (- 1/2 cos 2x) - { -1 (-1/4 cos 2x)} - cos 2x + c
= 1/2 x cos 2x - 1/4 cos 2x - cos 2x + c
= 1/4 ( 2x - 1 - 4 ) cos 2x + c
= 1/4 (2x - 5) cos 2x + c
6. carilah integral persial dari integral x^7 e^2x dx
Jawab:
Jawaban ada di gambar ya
Penjelasan dengan langkah-langkah:
7. minta tolong dong kak cara nentuin integral dari ∫▒e^2x/(2+e^2x ) dx gimana ya ?
Penjelasan dengan langkah-langkah:
[tex]\int \frac{ {e}^{2x} }{2 + {e}^{2x} } dx[/tex]
misal
[tex]u = 2 + {e}^{2x} \\ du = 2 {e}^{2x}dx \\ dx = \frac{du}{2 {e}^{2x} } [/tex]
[tex] =\int \frac{ {e}^{2x} }{u} \frac{du}{2 {e}^{2x} } \\ =\int \frac{1}{2u} du \\ = \frac{1}{2} ln \: u + c \\ = \frac{1}{2} ln \: (2+ {e}^{2x}) + c [/tex]
8. Tentukan nilai dari integral : integral 2x+2 cos 2x dx !
Integral hasil : x²+ sin 2x
9. cara mengintegralkan e^(2x-4)
∫e^2x-4 dx=1/2e^2x-4 +C
Dengan:
u = 2x - 4
du = 2 dx
Maka:
∫ e^(2x-4) dx
= 1/2 ∫ e^(2x-4) (2 dx)
= 1/2 ∫ e^u du
= 1/2 e^u + C
Substitusikan kembali:
= 1/2 e^(2x-4) + C
10. 1) integral cos (5x) + 1/x dx 2) integral e^2x - 3x + 1 dx 3) integral 3^2x+5 - 1/2x dx
jawab
1)
∫ (cos 5x + 1/x) dx
= ∫ (cos 5x + x⁻¹) dx
= 1/5 sin 5x + ln x + c
2)
∫ (e²ˣ - 3x + 1) dx
= 1/2 e²ˣ - 3/2 x² + x + c
3)
∫(3²ˣ⁺⁵ -1/2 x) dx
= (3²ˣ⁺⁵)/²ln 3 - 1/4 x² + C
11. integral parsial integral 2x pangkat 2 cos 2x dx
Jawab:
[tex]\displaystyle \int2x^2\cos2x\,dx=2x^2\cdot\frac12\sin2x-\int4x\cdot\frac12\sin2x\,dx\\\int2x^2\cos2x\,dx=x^2\sin2x-\int2x\sin2x\,dx\\\int2x^2\cos2x\,dx=x^2\sin2x-\left(2x\cdot\left(-\frac12\cos2x\right)-\int2\cdot\left(-\frac12\cos2x\right)\,dx\right)\\\int2x^2\cos2x\,dx=x^2\sin2x-\left(-x\cos2x+\int\cos2x\,dx\right)\\\int2x^2\cos2x\,dx=x^2\sin2x+x\cos2x-\int\cos2x\,dx\\\int2x^2\cos2x\,dx=x^2\sin2x+x\cos2x-\frac12\sin2x+C[/tex]
Beberapa konsep yang dipakai:
[tex]\displaystyle\triangleright~\int u\,dv=uv-\int v\,du\\\triangleright~\int\sin ax\,dx=-\frac1a\cos ax+C\\\triangleright~\int\cos ax\,dx=\frac1a\sin ax+C\\\triangleright~\frac{d}{dx}(ax^n)=anx^{n-1}[/tex]
12. mohon bantuannya1. integral x² e² dx2. integral e^x (x+1)² dx3. integral xe^3x dx4. integral 8x (4x³+12) dx5. integral 2x (5x²-8) dx
Hai, salam kenal.*^▁^*
Jawaban ada pada gambar!
13. Integral e pangkat x sin 2x dx
penyelesaian terlampir menggunakan rumus integral parsial.
14. Integral dr e^ 2x / akar e^2x - 1
integral substitusi :
misal u = e^(2x) - 1
du = 2e^(2x) dx
1/2 du = e^(2x) dx
so,
integral di atas dpt menjadi :
int 1/2 du/u = 1/2 ln(u) + c = 1/2 ln(e^(2x) - 1) + c
15. integral dari e^-3x cos 2x dx =
semoga membantu.....
16. 1) integral cos (5x)+1/x dx 2) integral e^2x-3x+1 dx 3) integral 3^2x+5-1/2x dx
jawab
semoga benar dengan maksud soal
1) ∫ (cos 5x + 1/x) dx
= ∫ (cos 5x + x⁻¹) dx
= 1/5 sin 5x + ln x + c
2) ∫ (e²ˣ - 3x + 1) dx
= 1/2 e²ˣ - 3/2 x² + x + c
3) ∫(3²ˣ⁺⁵ -1/2 x) dx
= (3²ˣ⁺⁵)/²ln 3 - 1/4 x² + C
17. tentukan a). integral e^7x dan b). integral 2x^5 sec^2 x - 2x^5 tg^2 x bantu jwab dong ka
Jawaban:
a) integral e^7x = 7e^7x + C
maaf saya tidak tau bagaimana mengerjakan yg b
18. Integral e^2x sin 3x
Berikut penjelasannya
∫ e^(2x)sin(3x) dx
let u = e^2x, dv = sin(3x) dx
du = 2e^2x dx, v = (-1/3)cos(3x)
uv - ∫ vdu
= (-(e^(2x)/3)cos(3x) + 2/3 ∫ (e^2x)(cos(3x) dx
let u = e^2x, dv = cos(3x) dx
du = 2e^2x dx, v = (1/3)sin(3x)
= (-(e^(2x)/3)cos(3x) + (2/3) [((e^2x)/3)sin(3x) + 2/3 ∫ (e^2x) sin(3x) dx]
= (-(e^(2x)/3)cos(3x) + (2/3)((e^2x)/3)sin(3x) + 4/9 ∫ (e^2x) sin(3x) dx
So we have:
∫ (e^2x) sin(3x) dx = (-(e^(2x)/3)cos(3x) + (2/3)((e^2x)/3)sin(3x) + 4/9 ∫ (e^2x) sin(3x) dx
Subtract 4/9 ∫ (e^2x) sin(3x) dx from both sides:
5/9 ∫ (e^2x) sin(3x) dx = (-(e^(2x)/3)cos(3x) + (2/3)((e^2x)/3)sin(3x)
Multiply both sides by 9/5:
∫ (e^2x) sin(3x) dx = (9/5)(-(e^(2x)/3)cos(3x) + (18/15)((e^2x)/3)sin(3x)
= (-3/5)(e^(2x)(cos(3x)) + (2/5)(e^2x)(sin(3x))INTEGRAL PARSIAL BERULANG
∫ uv dv = uv - ∫ v du
∫ e^(2x) sin 3x dx
u = e^(2x) → du = 2e^(2x) dx
dv = sin 3x → v = ∫ sin 3x dx = -1/3 cos 3x
∫ e^(2x) sin 3x dx = e^(2x) (-1/3 cos 3x) - ∫ (-1/3 cos 3x) 2e^(2x) dx
= -1/3 e^(2x) cos 3x - (-2/3 ∫ e^(2x) cos 3x dx)
u = e^(2x) → du = 2e^(2x) dx
dv = cos 3x → v = 1/3 sin 3x
∫ e^(2x) sin 3x dx = -1/3 e^(2x) cos 3x - {-2/3 [e^(2x) (1/3 sin 3x) - ∫ (1/3 sin 3x) 2e^(2x) dx]}
∫ e^(2x) sin 3x dx = -1/3 e^(2x) cos 3x - {-2/3[1/3 e^(2x) sin 3x - 2/3 ∫ e^(2x) sin 3x dx]}
∫ e^(2x) sin 3x dx = -1/3 e^(2x) cos 3x + 2/9 e^(2x) sin 3x - 4/9 e^(2x) sin 3x dx]
13/9 ∫ e^(2x) sin 3x dx = -1/3 e^(2x) cos 3x + 2/9 e^(2x) sin 3x
∫ e^(2x) sin 3x dx = -3/13 e^(2x) cos 3x + 2/13 e^(2x) cos 3x
∫ e^(2x) sin 3x dx = -1/13 e^(2x) [3 cos 3x - 2 sin 3x] + C
19. integral -5 e^2x+4 dx
Jawab:
-x × (5e² - 4d)
Penjelasan dengan langkah-langkah:
-5 e^2x+4 dx
= -x × (5e² - 4d)
makasih
semoga membantu
20. Integral of x(2x+3)^99 dx
let u = 2x + 3
2x = u - 3
x = (u - 3)/2
du = 2 dx
dx = du/2
∫x(2x + 3)^99 dx = ∫(u - 3)/2 * u^99 * du/2
= 1/4 ∫(u - 3)u^99 du
= 1/4 ∫u^100 - 3u^99 du
= 1/4((1/101)u^101 - (3/100)u^100) + C
= (1/404)(2x + 3)^101 - (3/400)(2x + 3)^100 + C
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