5.3 The Fundamental Theorem of Calculus/27: Difference between revisions
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\begin{align} | \begin{align} | ||
\int_0^2 x(2+x^5)\,dx = \int_0^2 (2x+x^6)\,dx | \int_0^2 x(2+x^5)\,dx &= \int_0^2 (2x+x^6)\,dx = \int_0^2 (2x+x^6)\,dx \\[2ex] | ||
&= \left(\frac{2x^{1+1}}{1+1}+\frac{x^{6+1}}{6+1}\right)\bigg|_{0}^{2}=\left(x^2+\frac{x^7}{7}\right)\bigg|_{0}^{2} \\[2ex] | &= \left(\frac{2x^{1+1}}{1+1}+\frac{x^{6+1}}{6+1}\right)\bigg|_{0}^{2}=\left(x^2+\frac{x^7}{7}\right)\bigg|_{0}^{2} \\[2ex] | ||
&= \left((2)^2-\frac{(2)^7}{7}\right) | &= \left((2)^2-\frac{(2)^7}{7}\right)-\left((0)^2+\frac{(0)^7}{7}\right) \\[2ex] | ||
&= \left[4+\frac{2^7}{7}\right]-[0] \\[2ex] | &= \left[4+\frac{2^7}{7}\right]-[0] \\[2ex] | ||
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\end{align} | \end{align} | ||
</math> | |||
Latest revision as of 21:17, 6 September 2022
![{\displaystyle {\begin{aligned}\int _{0}^{2}x(2+x^{5})\,dx&=\int _{0}^{2}(2x+x^{6})\,dx=\int _{0}^{2}(2x+x^{6})\,dx\\[2ex]&=\left({\frac {2x^{1+1}}{1+1}}+{\frac {x^{6+1}}{6+1}}\right){\bigg |}_{0}^{2}=\left(x^{2}+{\frac {x^{7}}{7}}\right){\bigg |}_{0}^{2}\\[2ex]&=\left((2)^{2}-{\frac {(2)^{7}}{7}}\right)-\left((0)^{2}+{\frac {(0)^{7}}{7}}\right)\\[2ex]&=\left[4+{\frac {2^{7}}{7}}\right]-[0]\\[2ex]&={\frac {156}{7}}\end{aligned}}}](https://en.wikipedia.org/api/rest_v1/media/math/render/svg/62167bee792832f664296ca9420fc370b9be2361)