7. Background information for those interested in the historical development (youmay skip this first paragraph and go straight to the question in the second para-graph if you are not interested in the history): The Cavalieri's quadrature formulagives an explicit solution to the integralx" dx. For the case of the parabola (n = 2), itwas proven by the ancient Greek mathematician Archimedes of Syracuse in the 3rd cen-tury BC in The Quadrature of the Parabola (written as a letter to his friend Dositheus).More precisely, Archimedes was using the method of exhaustion to find the area inside theparabola rather then the area under the graph. In the 11th century, Islamic mathematicianIbn al-Haytham computed the integrals of cubes (n = 3) and quartics (n = 4) using themethod of mathematicial induction. In the 17th century, an Italian Jesuit mathematicianBonaventura Cavalieri used the method of indivisibles and computed the cases for integersn from 1 up to 9. The French lawyer/mathematician Pierre de Fermat had also obtainedthe areas of parabolas and hyperbolas of any order. By using an ingenious trick, Fermatwas able to reduce the evaluation of the integral of general power to the sum of geometricseries. The trick, however, was not considered systematic nor explicit. In 1656, an En-glish mathematician John Wallis published his work Arithmetica Infinitorum in which theformula works for fractional and negative powers, although the case of the quadrature ofthe hyperbola (n = -1) was then interpreted incorrectly.VāQuestion: Compute the integral |x" dx in terms of a, assuming n is a positive inte-ger. Without performing further integration, findVx dx in terms of a.

Question

7. Background information for those interested in the historical development (youmay skip this first paragraph and go straight to the question in the second para-graph if you are not interested in the history): The Cavalieri's quadrature formulagives an explicit solution to the integralx&#34; dx. For the case of the parabola (n = 2), itwas proven by the ancient Greek mathematician Archimedes of Syracuse in the 3rd cen-tury BC in The Quadrature of the Parabola (written as a letter to his friend Dositheus).More precisely, Archimedes was using the method of exhaustion to find the area inside theparabola rather then the area under the graph. In the 11th century, Islamic mathematicianIbn al-Haytham computed the integrals of cubes (n = 3) and quartics (n = 4) using themethod of mathematicial induction. In the 17th century, an Italian Jesuit mathematicianBonaventura Cavalieri used the method of indivisibles and computed the cases for integersn from 1 up to 9. The French lawyer/mathematician Pierre de Fermat had also obtainedthe areas of parabolas and hyperbolas of any order. By using an ingenious trick, Fermatwas able to reduce the evaluation of the integral of general power to the sum of geometricseries. The trick, however, was not considered systematic nor explicit. In 1656, an En-glish mathematician John Wallis published his work Arithmetica Infinitorum in which theformula works for fractional and negative powers, although the case of the quadrature ofthe hyperbola (n = -1) was then interpreted incorrectly.VāQuestion: Compute the integral |x&#34; dx in terms of a, assuming n is a positive inte-ger. Without performing further integration, findVx dx in terms of a.

Accepted Answer

Answered: Image /qna-images/answer/7faaeb67-8bbe-4d3e-835e-a7101bf9d4eb.jpg

Va Question: Compute the integral / x" d.x in terms of a, assuming n is a positive inte- ger. Without performing further integration, find Vx dx in terms of a.