Resources for Reverse Chain Rule
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Questions
19
With Worked SolutionClick Here -
Video Tutorials
1
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HSC Questions
8
With Worked SolutionClick Here
Reverse Chain Rule Theory
![\(\displaystyle \int f^{\prime}(x)(f(x))^n d x=\dfrac{[f(x)]^{n+1}}{n+1}+c\)\\ 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}\\ %24634 Find \(\displaystyle \int {{{(1 - 2x)}^4}\,\,dx} \)\\ \textbf{Example 1 solution}\\ $\begin{aligned} \int(1-2 x)^{4} d x &=\frac{(1-2 x)^{5}}{5} \times-\frac{1}{2}+c \\ &=\frac{-1}{10}(1-2 x)^{5}+c \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ %11027 Evaluate \(\displaystyle \int\limits_0^1 {\dfrac{{3{x^2}}}{{\sqrt {1 + {x^3}} }}} \,dx\)\\ \textbf{Example 2 solution}\\ $\begin{aligned} I=\int_{0}^{1} 3 x^{2}\left(1+x^{3}\right)^{-\frac{1}{2}} d x &=\left[\frac{\left(1+x^{3}\right)^{\frac{1}{2}}}{\frac{1}{2}}\right]_{0}^{1} \\ &=2\left[\sqrt{1+x^{3}}\right]_{0}^{1} \\ &=2[\sqrt{2}-1] \end{aligned}$\\ \end{multicols}](/media/jz3dtsmk/4873.png)