# Integral Bentuk Dasar

## Konsep Dasar

A. $$\int x^n \: dx = \dfrac {1}{n + 1} \: x^{n + 1} + C$$

Contoh 1

\begin{equation*}
\begin{split}
\int x^4 \: dx & = \frac {1}{4 + 1} \: x^{4 + 1} + C \\\\
\int x^4 \: dx & = \tfrac {1}{5} x^{5} + C
\end{split}
\end{equation*}

Contoh 2

\begin{equation*}
\begin{split}
\int 3x^8 \: dx & = 3 \:.\: \frac {1}{8 + 1} \: x^{8 + 1} + C \\\\
\int 3x^8 \: dx & = \tfrac {1}{3} x^{9} + C
\end{split}
\end{equation*}

Contoh 3

\begin{equation*}
\begin{split}
\int \sqrt{x} \: dx & = \int x^{\frac 12} \: dx \\\\
\int \sqrt{x} \: dx & = \frac {1}{\frac 12 + 1} \: x^{\frac 12 + 1} + C \\\\
\int \sqrt{x} \: dx & = \tfrac {2}{3} x^{1 \frac 12} + C \\\\
\int \sqrt{x} \: dx & = \tfrac {2}{3} x \sqrt{x} + C \\\\
\end{split}
\end{equation*}

Contoh 4

\begin{equation*}
\begin{split}
\int \frac {1}{x^3} \: dx & = \int x^{-3} \: dx \\\\
\int \frac {1}{x^3} \: dx & = \frac {1}{-3 + 1} \: x^{-3 + 1} + C \\\\
\int \frac {1}{x^3} \: dx & = -\tfrac {1}{2} x^{-2} + C
\end{split}
\end{equation*}

B. $$\int x^{-1} \: dx = \ln |x| + C$$

Contoh 1

\begin{equation*}
\begin{split}
\int \frac 2x \: dx & = 2 x^{-1} \: dx \\\\
\int \frac 2x \: dx & = 2 \ln |x| + C
\end{split}
\end{equation*}

C. $$\int k \: dx = kx + C$$, k adalah konstanta

Contoh 1

\begin{equation*}
\begin{split}
\int 4 \: dx & = 4x + C
\end{split}
\end{equation*}

D. $$\int_a^b f(x) = F(b) - F(a)$$

Contoh 1

\begin{equation*}
\begin{split}
\int_1^4 3x^2 & = \left[3 \;.\: \frac {1}{2 + 1} \: x^{2 + 1}\right]_1^4  \\\\
\int_1^4 3x^2 & = \left[x^3\right]_1^4  \\\\
\int_1^4 3x^2 & = 4^3 - 1^3 \\\\
\int_1^4 3x^2 & = 63
\end{split}
\end{equation*}