find-a-formula-for-xkarctanx-dx-int-xk-arctanx-dx-where-k-k-is-a-positive-intege

Question

Find a formula for ∫xkarctan⁡(x) dx \int x^k \arctan(x) \, dx , where k k  is a positive integer.

Accepted Answer

xk+1k+1arctan⁡(x)−1k+1∫xk+11+x2 dx+C\frac{$$x^{k+1}$$}{k+1}\arctan(x)-\frac{1}{k+1}\int\frac{$$x^{k+1}$$}{1 + $$x^2$$}\,dx+C

Answer

xk+1k+1arctan⁡(x)−∫xk+11+x2 dx+C\frac{$$x^{k+1}$$}{k+1}\arctan(x)-\int\frac{$$x^{k+1}$$}{1 + $$x^2$$}\,dx+C

Answer

xk+1arctan⁡(x)−1k+1∫xk+11+x2 $$dx+Cx^{k+1}$$\arctan(x)-\frac{1}{k+1}\int\frac{$$x^{k+1}$$}{1 + $$x^2$$}\,dx+C

Answer

xk+1arctan⁡(x)−∫xk+11+x2 $$dx+Cx^{k+1}$$\arctan(x)-\int\frac{$$x^{k+1}$$}{1 + $$x^2$$}\,dx+C

Find a formula for ∫xkarctan(x)dx \(\int\) x^k \(\arctan\)(x) \, dx , where k k is a positive integer.