Question

17 Sampai 20 Tentang Integral​

Answer 1

Jawaban:

17.

mis: 2t² + 5 = x

dx/dt = 4t

dt = dx/4t

[tex]∫ \frac{6t}{2 {t}^{2} + 5} \: dt[/tex]

[tex] = ∫ \frac{6t}{x} \times \frac{dx}{4t} [/tex]

[tex] = ∫ \frac{6}{4} \times \frac{1}{x} \: dx[/tex]

[tex] = \frac{3}{2} ∫ \frac{1}{x} \: dx[/tex]

[tex] = \frac{3}{2} ln \: x + c[/tex]

[tex] = \frac{3}{2} ln(2 {t}^{2} + 5)+c[/tex]

18.

mis: 4 - e^3x = u

du/dx = -3e^(3x)

dx = du/(-3e^(3x))

[tex]∫\frac{{e}^{3x}}{4-{e}^{3x}} \: dx[/tex]

[tex] = ∫\frac{{e}^{3x}}{u} \times \frac{du}{-3{e}^{3x}}[/tex]

[tex] = ∫-\frac{1}{u} \times \frac{du}{3}[/tex]

[tex] = -\frac{1}{3}ln \: u + c[/tex]

[tex] = -\frac{1}{3}ln(4-{e}^{3x})+ c[/tex]

19.

mis: e^x + 1 = u

du/dx = e^x

dx = du/e^x

[tex]∫\frac{{2e}^{x}}{{e}^{x}+1} \: dx[/tex]

[tex] = ∫\frac{{2e}^{x}}{u} \times \frac{du}{{e}^{x}}[/tex]

[tex] = ∫\frac{2}{u} \times \: du[/tex]

[tex] = 2ln(u) + c[/tex]

[tex] = 2ln({e}^{x}+1) + c[/tex]

batas -ln(2) s/d 0 :

[tex] = 2ln({e}^{0}+1) - 2ln({e}^{-ln(2)}+1)[/tex]

[tex] = 2ln(1+1) - 2ln(0,5+1)[/tex]

[tex] = 2(ln(2) - ln(1,5))[/tex]

[tex] = 2ln(\frac{4}{3})[/tex]

20.

mis: ln(x) = u

du/dx = 1/x

dx = x•du

[tex]∫\frac{3}{xln(x)} \: dx[/tex]

[tex] = ∫\frac{3}{x} \times \frac{1}{u} \times x \: du[/tex]

[tex] = 3∫\frac{1}{u} \: du[/tex]

[tex] = 3ln(u)[/tex]

[tex] = 3ln(ln(x))[/tex]