Sep
10
2012

Rumus-rumus Integral

Berikut ini adalah kumpulan rumus-rumus yang biasa dipakai untuk menyelesaikan soal-soal integral.

Integral Fungsi Aljabar

    \begin{align*}   \boxed{\int k \: \mathrm{d}x = kx + C} \end{align*}

    \begin{align*}   \boxed{\int x^n \: \mathrm{d}x = \frac{x^{n+1}}{n+1} + C \quad \text{Untuk } x \neq -1} \end{align*}

    \begin{align*}   \boxed{\int (ax+b)^n \: \mathrm{d}x = \frac{(ax+b)^{n+1}}{a(n+1)} + C \quad \text{Untuk } x \neq -1} \end{align*}

    \begin{align*}   \boxed{\int \frac{1}{x} \: \mathrm{d}x = \ln |x| + C} \end{align*}

    \begin{align*}   \boxed{\int \frac{c}{ax+b} \: \mathrm{d}x = \frac{c}{a} \ln |ax+b| + C} \end{align*}

Integral Fungsi Eksponen

    \begin{align*}   \boxed{\int e^x \: \mathrm{d}x = e^x + C} \end{align*}

    \begin{align*}   \boxed{\int a^x \: \mathrm{d}x = \frac{a^x}{\ln a} + C} \end{align*}

Integral Fungsi Logaritma

    \begin{align*}   \boxed{\int \ln x \: \mathrm{d}x = x \ln x - x + C} \end{align*}

    \begin{align*}   \boxed{\int \log_a x \: \mathrm{d}x = x \log_a x - \frac{x}{\ln a} + C} \end{align*}

Integral Fungsi Trigonometri

    \begin{align*}   \boxed{\int \sin x \: \mathrm{d}x = -\cos x + C} \end{align*}

    \begin{align*}   \boxed{\int \cos x \: \mathrm{d}x = \sin x + C} \end{align*}

    \begin{align*}   \boxed{\int \tan x \: \mathrm{d}x = -\ln|\cos x| + C = \ln |\sec x| + C} \end{align*}

    \begin{align*}   \boxed{\int \cot x \: \mathrm{d}x = \ln |\sin x| + C} \end{align*}

    \begin{align*}   \boxed{\int \sec x \: \mathrm{d}x = \ln |\sec x + \tan x| + C} \end{align*}

    \begin{align*}   \boxed{\int \csc x \: \mathrm{d}x = -\ln |\csc x + \cot x| + C} \end{align*}

    \begin{align*}   \boxed{\int \sec^2 x \: \mathrm{d}x = \tan x + C} \end{align*}

    \begin{align*}   \boxed{\int \csc^2 x \: \mathrm{d}x = -\cot x + C} \end{align*}

    \begin{align*}   \boxed{\int \sec x \tan x\: \mathrm{d}x = \sec x + C} \end{align*}

    \begin{align*}   \boxed{\int \csc x \cot x \: \mathrm{d}x = -\csc x + C} \end{align*}

Integral Fungsi Invers Trigonometri

    \begin{align*}   \boxed{\int \frac{\mathrm{d}x}{\sqrt{1-x^2}} = \arcsin x + C} \end{align*}

    \begin{align*}   \boxed{\int \frac{\mathrm{d}x}{x\sqrt{x^2-1}} = \arcsec x + C} \end{align*}

    \begin{align*}   \boxed{\int \frac{\mathrm{d}x}{1+x^2} = \arctan x + C} \end{align*}

    \begin{align*}   \boxed{\int \arcsin x \: \mathrm{d}x = x \arcsin x + \sqrt{1-x^2} + C} \end{align*}

    \begin{align*}   \boxed{\int \arccos x \: \mathrm{d}x = x \arccos x - \sqrt{1-x^2} + C} \end{align*}

    \begin{align*}   \boxed{\int \arctan x \: \mathrm{d}x = x \arctan x - \frac{1}{2} \ln |1+x^2| + C} \end{align*}

    \begin{align*}   \boxed{\int \operatorname{arccot} x \: \mathrm{d}x = x \operatorname{arccot} x + \frac{1}{2} \ln |1+x^2| + C} \end{align*}

Integral Parsial

    \begin{align*}   \boxed{\int u \: \mathrm{d}v = u \cdot v - \int v \: \mathrm{d}u} \end{align*}

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