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NSW Y12 Maths - Advanced Integration Indefinite Integral

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Indefinite Integral Theory

\(\displaystyle \int x^n \,d x=\dfrac{x^{n+1}}{n+1}+c\)\\ Special Rule\\ \(\displaystyle \int(a x+b)^n \,d x=\dfrac{(a x+b)^{n+1}}{a(n+1)}+c\)\\ \begin{multicols}{2} \textbf{Example 1}\\ \(\displaystyle \int \sqrt{x}\, d x\)\\ \textbf{Example 1 solution}\\ $\begin{aligned} \displaystyle \int \sqrt{x} d x & =\displaystyle \int x^{\frac{1}{2}}\, d x \\ & =\frac{x^{\frac{3}{2}}}{\frac{3}{2}}+c \\ & =\frac{2}{3} x \sqrt{x}+c \end{aligned}$\\ \textbf{Example 2}\\ \(\displaystyle \int \dfrac{1}{3 x^2}\, d x\)\\ \textbf{Example 2 solution}\\ $\begin{aligned} \displaystyle \int \frac{1}{3 x^2} \,d x & =\frac{1}{3} \displaystyle \int x^{-2} \,d x \\ & =\frac{1}{3} \times \frac{x^{-1}}{-1}+c \\ & =\frac{-1}{3 x}+c\end{aligned}$\\ \columnbreak \textbf{Example 3}\\ \(\displaystyle \int \dfrac{x^4+1}{x^2} d x\)\\ \textbf{Example 3 solution}\\ $\begin{aligned} \displaystyle \int \frac{x^4+1}{x^2} \,d x & =\displaystyle \int \frac{x^4}{x^2}+\frac{1}{x^2} \,d x \\ & =\displaystyle \int x^2+x^{-2} \,d x \\ & =\frac{x^3}{3}+\frac{x^{-1}}{-1}+c \\ & =\frac{1}{3} x^3-\frac{1}{x}+c \end{aligned}$\\ \textbf{Example 4}\\ %30948 Find \(\displaystyle \int (1 - 3x)^5 \, dx\)\\ \textbf{Example 4 solution}\\ $\begin{aligned} \int (ax + b)^n \, dx &= \frac{(ax + b)^{n + 1}}{a(n+1)} + C\\\therefore \int (1 - 3x)^5 \, dx &= \frac{(1 - 3x)^6}{-3\times 6} +C\\&= -\frac{(1-3x)^6}{18} + C \end{aligned}$\\ \end{multicols}

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  • Indefinite Integral - Video - Intro to Integration and integrating polynomials

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