$\text { Use integration tables to find the integral } \int \sqrt [ 4 ] { x } \arctan x ^ { 5 / 4 } d x \text {. }$
A) $x ^ { 5 / 4 } \arctan \left( x ^ { 5 / 4 } \right) + \ln \sqrt { 1 + x ^ { 5 / 2 } } + C$
B) $\frac { 5 } { 4 } \left[ x ^ { 5 / 4 } \arctan \left( x ^ { 5 / 4 } \right) + \ln \sqrt { 1 + x ^ { 5 / 2 } } \right] + C$
C) $\frac { 1 } { 9 } \left( x ^ { 5 / 4 } \arctan \left( x ^ { 5 / 4 } \right) - \ln \sqrt { 1 + x ^ { 5 / 4 } } \right) + C$
D) $\frac { 4 } { 5 } \left( x ^ { 5 / 4 } \arctan \left( x ^ { 5 / 4 } \right) - \ln \sqrt { 1 + x ^ { 5 / 2 } } \right) + C$
E) $x ^ { 5 / 4 } \arctan \left( x ^ { 5 / 4 } \right) - \ln \sqrt { 1 + x ^ { 5 / 4 } } + C$

Question

Quizplus · Accepted Answer

The integral involves a combination of a function and its derivative, which suggests the use of integration by parts. The correct answer must balance the derivative of $x^{5/4}$ and the integral of $\arctan(x^{5/4})$, along with the adjustment of constants to match the integration by parts formula. Option D correctly reflects the structure expected from integrating a product of $x^{n}$ and $\arctan(x^{n})$, including the necessary constant factor to balance the differentiation of $x^{5/4}$ and the integration of $\arctan(x^{5/4})$, along with the logarithmic term that typically results from integrating an arctan function.

Use integration tables to find the integral ∫x4arctan⁡x5/4dx. \text { Use integration tables to find the integral } \int \sqrt [ 4 ] { x } \arctan x ^ { 5 / 4 } d x \text {. } Use integration tables to find the integral ∫4x​arctanx5/4dx.

$\text { Use integration tables to find the integral } \int \sqrt [ 4 ] { x } \arctan x ^ { 5 / 4 } d x \text {. }$