Resources for Definite Integral
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Questions
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Video Tutorials
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HSC Questions
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Definite Integral Theory
![$\begin{aligned} \displaystyle \int_a^b x^n \,d x & =\left[\frac{x^{n+1}}{n+1}\right]_a^b \\ & =\frac{1}{n+1}\left[b^{n+1}-a^{n+1}\right] \end{aligned}$\\ \begin{multicols}{2} \textbf{Example 1}\\ %24607 Find \(\displaystyle \int\limits_1^4 {x - \dfrac{1}{{\sqrt x }}\,\,dx} \) \\ \textbf{Example 1 solution}\\ $\begin{aligned} \int_{1}^{4} x-x^{-\frac{1}{2}} d x &=\left[\frac{x^{2}}{2}-2 \sqrt{x}\right]_{1}^{4} \\ &=\left[(2-4)-\left(\frac{1}{2}-2\right)\right] \\ &=-\frac{1}{2} \end{aligned}$\\ \columnbreak \textbf{Example 2}\\ %30947 Evaluate the following integral \(\displaystyle \int_1^3 \frac{1}{x^3} \, dx \)\\ \textbf{Example 2 solution}\\ $\begin{aligned} \int_1^3 \frac{1}{x^3} \, dx &= \int_1^3 x^{-3} \, dx\\&= \left[\frac{x^{-2}}{-2}\right]_1^3\\&= -\frac{1}{2}\left[\frac{1}{x^2}\right]_1^3\\&= -\frac{1}{2}\left[ \frac{1}{9} - 1 \right]\\&= \frac{4}{9} \end{aligned}$\\ \end{multicols}](/media/1rmcwp41/4870.png)