∫1/(1+cosx) dx dan ∫(x+2)/√(1-3x^2 ) dx yang bisa tolong bantu yaa.... Question from @hanamutiadewi

September 2018 1 72 Report

∫1/(1+cosx) dx

dan

∫(x+2)/√(1-3x^2 ) dx

yang bisa tolong bantu yaa

Takamori37 Atas:
$\displaystyle \int\frac{1}{1+\cos x}\,dx=\int\frac{dx}{1+2\cos^2(\frac{x}{2})-1}=\int\frac{dx}{2\cos^2(\frac{x}{2})} \\\\ \int \frac{\sec^2(\frac{x}{2})}{2}\, dx \\\\ u=\frac{x}{2}~~~~~~~~du=\frac{1}{2}\, dx \\ \text{Integralnya menjadi:} \\ \int\sec^2u\, du=\tan u+C=\boxed{\tan\frac{x}{2}+C}$

Bawah:
$\displaystyle \int\frac{x+2}{\sqrt{1-3x^2}}\, dx=\int\frac{x}{\sqrt{1-3x^2}}\, dx+\int\frac{2}{\sqrt{1-3x^2}}\, dx$

$\displaystyle \text{Untuk $\int\frac{x}{\sqrt{1-3x^2}}\, dx$} \\\\ u=1-3x^2~~~~~du=-6x\, dx~~~\to~~~x\, dx=-\frac{1}{6}\, du \\\\ \int-\frac{du}{6\sqrt{u}}=-\frac{1}{6}\int u^{-1/2}\,du=-\frac{1}{6}\times\frac{u^{1/2}}{1/2}+C=-\frac{\sqrt{u}}{3}+C \\ \text{Maka:} \\ \int\frac{x}{\sqrt{1-3x^2}}\, dx=-\frac{\sqrt{1-3x^2}}{3}+C$

$\displaystyle \text{Untuk $\int\frac{2}{\sqrt{1-3x^2}}\, dx$} \\ \text{Ada rumus:} \\ \int\frac{1}{\sqrt{a^2-bx^2}}\, dx=\int\frac{1}{\sqrt{b}\sqrt{\frac{a^2}{b}-x^2}}\, dx=\frac{1}{\sqrt{b}}\int\frac{1}{\sqrt{\frac{a^2}{b}-x^2}}\, dx \\\\ \int\frac{1}{\sqrt{a^2-bx^2}}\, dx=\frac{1}{\sqrt{b}}\sin^{-1}\frac{x}{a/\sqrt{b}}+C=\frac{1}{\sqrt{b}}\sin^{-1}\frac{\sqrt{b}x}{a}+C$

$\displaystyle \\ \text{Dengan koefisien 2, $a^2=1,\,b=\sqrt{3}$. Akan menjadi:} \\ \int\frac{2}{\sqrt{1-3x^2}}\, dx=\frac{2}{\sqrt{3}}\sin^{-1}\frac{\sqrt{3}x}{1}+C=\frac{2\sin^{-1}(\sqrt{3}x)}{\sqrt{3}}+C$

Gabungkan dua integral tersebut:
$\displaystyle \int\frac{x+2}{\sqrt{1-3x^2}}\, dx=\int\frac{x}{\sqrt{1-3x^2}}\, dx+\int\frac{2}{\sqrt{1-3x^2}}\, dx \\\\ \int\frac{x+2}{\sqrt{1-3x^2}}\, dx=-\frac{\sqrt{1-3x^2}}{3}+\frac{2\sin^{-1}(\sqrt{3}x)}{\sqrt{3}}+C \\\\ \int\frac{x+2}{\sqrt{1-3x^2}}\, dx=\frac{2\sin^{-1}(\sqrt{3}x)}{\sqrt{3}}-\frac{\sqrt{1-3x^2}}{3}+C \\\\ \boxed{\int\frac{x+2}{\sqrt{1-3x^2}}\, dx=\frac{1}{3}\left(2\sqrt{3}\sin^{-1}(\sqrt{3}x)-\sqrt{1-3x^2}\right)+C}$

2 votes Thanks 1

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