Final answer to the problem
Step-by-step Solution
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Divide $x^7+16$ by $x^2+5x+6$
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$\begin{array}{l}\phantom{\phantom{;}x^{2}+5x\phantom{;}+6;}{\phantom{;}x^{5}-5x^{4}+19x^{3}-65x^{2}+211x\phantom{;}-665\phantom{;}\phantom{;}}\\\phantom{;}x^{2}+5x\phantom{;}+6\overline{\smash{)}\phantom{;}x^{7}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+16\phantom{;}\phantom{;}}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+6;}\underline{-x^{7}-5x^{6}-6x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{-x^{7}-5x^{6}-6x^{5};}-5x^{6}-6x^{5}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+16\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+6-;x^n;}\underline{\phantom{;}5x^{6}+25x^{5}+30x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;\phantom{;}5x^{6}+25x^{5}+30x^{4}-;x^n;}\phantom{;}19x^{5}+30x^{4}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}+16\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+6-;x^n-;x^n;}\underline{-19x^{5}-95x^{4}-114x^{3}\phantom{-;x^n}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;-19x^{5}-95x^{4}-114x^{3}-;x^n-;x^n;}-65x^{4}-114x^{3}\phantom{-;x^n}\phantom{-;x^n}+16\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+6-;x^n-;x^n-;x^n;}\underline{\phantom{;}65x^{4}+325x^{3}+390x^{2}\phantom{-;x^n}\phantom{-;x^n}}\\\phantom{;;;\phantom{;}65x^{4}+325x^{3}+390x^{2}-;x^n-;x^n-;x^n;}\phantom{;}211x^{3}+390x^{2}\phantom{-;x^n}+16\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+6-;x^n-;x^n-;x^n-;x^n;}\underline{-211x^{3}-1055x^{2}-1266x\phantom{;}\phantom{-;x^n}}\\\phantom{;;;;-211x^{3}-1055x^{2}-1266x\phantom{;}-;x^n-;x^n-;x^n-;x^n;}-665x^{2}-1266x\phantom{;}+16\phantom{;}\phantom{;}\\\phantom{\phantom{;}x^{2}+5x\phantom{;}+6-;x^n-;x^n-;x^n-;x^n-;x^n;}\underline{\phantom{;}665x^{2}+3325x\phantom{;}+3990\phantom{;}\phantom{;}}\\\phantom{;;;;;\phantom{;}665x^{2}+3325x\phantom{;}+3990\phantom{;}\phantom{;}-;x^n-;x^n-;x^n-;x^n-;x^n;}\phantom{;}2059x\phantom{;}+4006\phantom{;}\phantom{;}\\\end{array}$
Learn how to solve problems step by step online. Integrate the function (x^7+16)/(x^2+5x+6) from 0 to 1. Divide x^7+16 by x^2+5x+6. Resulting polynomial. Expand the integral \int_{0}^{1}\left(x^{5}-5x^{4}+19x^{3}-65x^{2}+211x-665+\frac{2059x+4006}{x^2+5x+6}\right)dx into 7 integrals using the sum rule for integrals, to then solve each integral separately. The integral \int_{0}^{1} x^{5}dx results in: \frac{1}{6}.