Answer to Question #202806 in Mechanics | Relativity for Ayush

Question

Derive the reduction formula

integeration(x²+a²)^n/2dx = x(x²+a²)^n/2÷n+1+ na²/n+1×integeration (x² +a²)^n/2-1dx

Use the formula to integrate integration(x²+a²)^5/2

Answer 1

$\int (x^2+a^2)^{\frac52}dx=\frac{x(x^2+a^2)^{\frac52}}{6}+\frac{5a^2}{6}\int (x^2+a^2)^{\frac32}dx=\frac{x(x^2+a^2)^{\frac52}}{6}+\frac{5a^2}{6}(\frac{x(x^2+a^2)^{\frac32}}{4}+\frac{3a^2}{4}\int (x^2+a^2)^{\frac12}dx)=$ $\frac{x(x^2+a^2)^{\frac52}}{6}+\frac{5a^2x(x^2+a^2)^{\frac32}}{24}+\frac{5a^4}{16}(x(x^2+a^2)^{\frac12}+a^2\ln(x+(x^2+a^2)^{\frac12}))+C.$