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1a)
![{\displaystyle \int 2x^{5}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/299ba9ff8ac8ceeb9ef1093ef8cf14e850b2d291)
- Using our rule: That
is equal to ![{\displaystyle y={\frac {x^{(n+1)}}{(n+1)}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/71b1fb16e652f18a8d4edccb6baaa0dd11e44210)
- We get:
![{\displaystyle y={\frac {2x^{6}}{6}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/8d934988a50b8f89740ec5d9ec4854c724b3d844)
b)
![{\displaystyle \int 7x^{6}+2x^{3}-x^{2}\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/19f572aad8153600772405b8c9e3f41e7eeda9ec)
- Again using our rule, we would get:
![{\displaystyle y=x^{7}+{\frac {x^{4}}{2}}-{\frac {x^{3}}{3}}+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4ca3a248ae51838aa52a72e70ea7485e1c2290fd)
2a)
given that the point
lies on the curve.
- Using our rule, the integral becomes
![{\displaystyle y={\frac {x^{2}}{2}}+5x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/68fcc554518727ef1cbb80f6bec2a5d1df3d1fc7)
- Now we can sub in our points
, So that:
![{\displaystyle 3={\frac {0^{2}}{2}}+5(0)+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/09a1c4b5a596c8cedcd807b6ec7ce9999872b565)
- Therefore C = 3
b)
![{\displaystyle \int 3x^{2}+7x+0.1\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b0ca7e409eb134e6fbaa3cdddcf308617631a72b)
- Evaulating this we get:
![{\displaystyle x^{3}+{\frac {7x^{2}}{2}}+0.1x+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/52bf62220e45c4d217951945d7e4a9fd6670b636)
- Given (2,2), subing these points in:
![{\displaystyle 2=2^{3}+{\frac {7(2^{2})}{2}}+0.2+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7f782e7c874b111076cd84a6309442da9fa00438)
![{\displaystyle 2=8+14+0.2+C}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7b0087dcd4140813b7f5ac75ad83f260cac5fbb2)
![{\displaystyle C=-20.2}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4528af30e460806f908bccdf6001218d29eb67e8)
3a)
![{\displaystyle \int _{0}^{2}x+1\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7d79c95b144c5ad39cb0cf51e9b3fab8aea167a9)
- Evaluating this we get:
![{\displaystyle {\Bigg \lfloor }{\frac {x^{2}}{2}}+x{\Bigg \rceil }_{0}^{2}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/531bfdf0cc1eb17b7a0df59cbd44bf55189c6fc2)
- Substituting in values we get:
![{\displaystyle {\Bigg \lfloor }\left({\frac {2^{2}}{2}}+2\right)-\left({\frac {0^{2}}{2}}+0\right){\Bigg \rceil }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/6fed2c4340e1a3264c710ba331e049aa74653ee5)
![{\displaystyle =4}](https://wikimedia.org/api/rest_v1/media/math/render/svg/2bbe2072f480789d021eb04e8c224613042ef6b0)
b)
![{\displaystyle \int _{-3}^{4.7}{\frac {1}{7}}x^{\frac {1}{3}}+1\,dx}](https://wikimedia.org/api/rest_v1/media/math/render/svg/b97d552142c0d0581ca7bbc37904c48bc35b646d)
- Evaluating this we get:
![{\displaystyle {\Bigg \lfloor }{\frac {3x^{\frac {4}{3}}}{28}}+x{\Bigg \rceil }_{-3}^{4.7}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/7a3a03daa40cdd88175097917c2f825ddec10b3c)
![{\displaystyle {\Bigg \lfloor }\left({\frac {3(4.7)^{\frac {4}{3}}}{28}}+4.7\right)-\left({\frac {3(-3)^{\frac {4}{3}}}{28}}-3\right){\Bigg \rceil }}](https://wikimedia.org/api/rest_v1/media/math/render/svg/85f24d0f48309de1dbcd0b8a36c30929efc9effd)
![{\displaystyle \approx 8.08}](https://wikimedia.org/api/rest_v1/media/math/render/svg/3468b0fb0d7a6fc370759a20bbd95f76f0baa399)
4)
- The question is simply to evaluate this definite integral:
![{\displaystyle {\begin{aligned}\int _{-2}^{0}{\bigg (}y=-x^{4}-{\frac {1}{2}}x^{3}+3x^{2}{\bigg )}\,dx&={\Bigg \lfloor }\left({\frac {-1}{5}}x^{5}-{\frac {1}{8}}x^{4}+x^{3}\right){\Bigg \rceil }_{-2}^{0}\\&=-{\bigg (}{\frac {-1}{5}}*-2^{5}-{\frac {1}{8}}*-2^{4}-2^{3}{\bigg )}\\&=3.6\end{aligned}}}](https://wikimedia.org/api/rest_v1/media/math/render/svg/4dfddeb806d596ae67123f7f5a4936964bdbde6b)